G.f. for Matrix Inverse of Triangle with Known G.F.

[Ralf Stephan]

> Now the $100,000 question arises: 
>  
> for a known g.f. of an invertible triangle T, what is the transform of
> variables x,y, 
> that converts the g.f. of T into the g.f.of the matrix inverse of T? 

One problem with that is that only ~3 per cent of all triangles/arrays have
known o.g.f.s (not only those for k-th row or column). As the list is short
and, without doubt, incomplete, I'll include it here for your perusal:

(127 lines)

%F A000027 a(n)=n. G.f.: x/(1-x)^2. E.g.f.: xe^x.
%F A001263 G.f.: (1+x(1-y)-sqrt(1-2x(1+y)+x^2(1-y)^2))/(2x)-1 = Sum_{n>0,k>0} a(n,k) x^n y^k.
%F A003506 G.f.: x*y/(1-x-y*x)^2. E.g.f: x*y*exp(x+x*y). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 01 2003
%F A003982 G.f.: 1/(1 - xy). E.g.f.: exp(xy).
%F A003991 G.f.: x * y / [ (1-x)^2 * (1-y)^2 ].
%F A007318 G.f.: 1/(1-y-xy)=Sum(C(n,k)x^k*y^n, n,k>=0); also g.f.: 1/(1-x-y)=Sum(C(n+k,k)x^k*y^n, n,k>=0). G.f. for row n: (1+x)^n = sum(k=0..n,C(n,k)x^k). G.f. for column n: x^n/(1-x)^n.
%F A008284 G.f.: A(x,y) = Product_{n>=1} 1/(1-x^n)^(P_n(y)/n), where P_n(y) = Sum_{d|n} eulerphi(n/d)*y^d. - Paul D Hanna (pauldhanna(AT)juno.com), Jul 13 2004
%F A008459 G.f.: 1/sqrt(1-2*y-2*x*y+y^2-2*x*y^2+x^2*y^2); g.f. for row n: (1-t)^n P_n[(1+t)/(1-t)] where the P_n's are the Legendre polynomials. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 23 2003
%F A009766 G.f.=C(tx)/[1-xC(tx)]=1+(1+t)x+(1+2t+2t^2)x^2+..., where C(x)=[1-sqrt(1-4x)]/(2x) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 18 2004
%F A010766 G.f.: 1/(1-x)*Sum_(k>=1} x^k/(1-y*x^k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 05 2004
%F A011117 G.f.=2/[1+uv-2v+sqrt(1-6uv+u^2v^2)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2003
%F A013580 G.f.: 1/(1-(1+y)*x)/(1-y*x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 12 2003
%F A016095 G.f.: 1/(1-x-y-(x+y)^2).
%F A026374 T(2n,k)=sum(3^(2j-k)*binomial(n,j)binomial(j,k-j), j=ceil(k/2)..k); T(2n+1,k)=T(2n,k-1)+T(2n,k). G.f.=(1+z+tz)/[1-(1+3t+t^2)z^2]=1+(1+t)z+(1+3t+t^2)z^2+... . Generating polynomial for row 2n is (1+3t+t^2)^n and for row 2n+1 it is (1+t)(1+3t+t^2)^n. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2004
%F A026807 G.f: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 22 2003
%F A026835 G.f: Sum_{k>=1} (y^k*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 25 2003
%F A029635 G.f.: (1+xy)/(1-x-xy). - Michael Somos, Jul 15 2003
%F A029653 G.f.: (1+x+y+xy)/(1-y-xy). - R. Stephan, May 17 2004
%F A033184 G.f.= txc/(1-txc), where c=(1-sqrt(1-4x))/(2x) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004
%F A033282 G.f. G=G(t,z) satisfies (1+t)G^2-z(1-z-2tz)G+tz^4=0.
%F A033820 G.f.: 2/(1-4*x*y+sqrt((1-4*x)*(1-4*x*y))). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 14 2003
%F A033877 G.f.: (1-x*y-(x^2*y^2-6*x*y+1)^(1/2))/(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 16 2003
%F A034807 G.f.: (2-x)/(1-x-x^2*y). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 31 2003
%F A034929 G.f.= 2(1+tz)/[1-2z+tz-2tz^2+sqrt(1-2tz-3t^2*z^2)].
%F A035607 G.f.: (1+x)/(1-x-x*y-x^2*y). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 02 2002
%F A036355 G.f.: 1/(1-(1+y)*x-(1+y^2)*x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 11 2003
%F A048887 G.f.: (1-z)/[1-2z+z^(t+1)].
%F A054106 G.f.: 1/(1-(1+y)*x)/(1+y*x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 12 2003
%F A055277 G.f. satisfies A(x,y)=xy+x*EULER(A(x,y))-x. Shifts up under EULER transform.
%F A055290 G.f.: A(x,y)=(1-x+x*y)*B(x,y)+(1/2)*(B(x^2,y^2)-B(x,y)^2). B(x,y): g.f. of A055277.
%F A055340 G.f. satisfies A(x,y)=xy+x*CIK(A(x,y))-x. Shifts up under CIK transform.
%F A055372 a(n,k)=2^(n-1)*C(n,k). G.f.: A(x,y)=(1-x-xy)/(1-2x-2xy).
%F A055898 G.f.: A(x,y)=(1/2x)((1-(4x/((1+x)(1+x-xy))))^(-1/2) - 1).
%F A055921 G.f.: A(x,y)=(1-x^2)*(1-y^2)/(1-x-y-xy)^2 * (1 - 16*x^2*y^2/((1-x^2)^2*(1-y^2)^2))^(1/4).
%F A059259 G.f.: 1/(1-x-x*y-y^2)
%F A059260 G.f.: 1/(1-y-x*y-x^2) = 1 + y + x^2 + xy + y^2 + 2x^2y + 2xy^2 + y^3 + ...
%F A059283 G.f. for T(n,k): ((1+2*w+w^2)*z^2+(-1-2*w-w^2)*z-w*(-3*w^2-6*w+1)^(1/2)+2*w)/(1+w)^2/((1+w)*z^2+(w-1)*z+w) (expand first as series in z, then in w).
%F A059473 G.f.: 1/(1-2*z-2*w-2*z*w).
%F A059474 G.f.: 1/(1-2*z-2*w+2*z*w).
%F A059576 To give a g.f. let the entries be relabeled U(0,0), U(1,0), U(0,1), U(2,0), U(1,1), U(0,2), U(3,0), ... Then g.f. = Sum_{n >= 0, k >= 0} U(n,k)*z^n*w^k = (1-z)*(1-w)/(1-2*w-2*z+2*z*w). Maple code gives explicit formula for U(n,k).
%F A060693 G.f.: (1-ty-sqrt((1-yt)^2-4y))/2.
%F A060920 G.f.: (1-x*(1+y))/(1-(3+2*y)*x+(1+y)^2*x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 11 2003
%F A060921 G.f.: 1/(1-(3+2*y)*x+(1+y)^2*x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 11 2003
%F A062110 G.f.: 1/(1-x(1-y)/(1-2y)) = Sum_{i,j} a(i,j)x^i*y^j.
%F A063007 G.f.=G(t,z)=1/sqrt(1-2z-4tz+z^2). Row generating polynomials=P_n(1+2z), i.e. T(n,k)=[z^k]P_n(1+2z), where P_n are the Legendre polynomials. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2004
%F A063967 G.f.: 1/(1-x*(1+x)*(1+y)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 11 2003
%F A064189 G.f. = M/(1-tzM), where M=1+zM+z^2M^2 is the g.f. of the Motzkin numbers (A001006). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 29 2004
%F A064861 G.f.: sum_{m=0}^infinity sum_{n=0}^infinity a_{m,n}t^m s^n=A(t,s)=(1+2t+s)/(1-2t^2-s^2-3st)
%F A064879 G.f.: (x^m)*(1+(1-2*m)*x*c(x*m^2))/(1-m*x*c(x*m^2))^2 = (x^m)*((2*m-1)*c(x*m^2)*(m*x)^2 +(1-m)*(1-m+(1-3*m)*x))/(1-m-m*x)^2, m >= 0. For m >= 1 also: (x^m)*c(x*m^2)*(2*m-1+c(x*m^2)*(m-1)^2)/(1+(m-1)*c(x*m^2))^2.
%F A065600 G.f. (1 - (1 - 4*x)^(1/2))/(x*(3 - y + (1 - 4*x)^(1/2)*(y-1))) = Sum_{n>=0,k>=0} T(n,k)x^n*y^k. - David Callan (callan(AT)stat.wisc.edu), Aug 17 2004
%F A065602 T(n,k)= sum((k-1+2j)*binomial(2n-k-1-2j,n-1)/(2n-k-1-2j),j=0..floor((n-k)/2)). G.f.=t^2*z^2*C/[(1-z^2*C^2)(1-tzC)], where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2004
%F A065941 G.f.: sum[n, sum[k, T(k,n)x^ky^n]] = (1+xy)/(1-y-x^2y^2). sum[n>=0, T(k,n)y^n] = y^k/(1-y)^[k/2]. - Ralf Stephan, May 17 2004
%F A066855 G.f.: Product_{m=1..infinity} (1-y*x^m)^(-A001055(m)). T(n,k) = Sum_{pi} Product_{m=1..n} binomial(p(m)+A001055(m)-1,p(m)), where pi runs through all nonnegative solutions of p(1)+2*p(2)+...+n*p(n)=n, p(1)+p(2)+...+p(n)=k.
%F A067804 G.f.: 1/sqrt((1-4*x)*(1-4*x*y)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 12 2003
%F A070543 G.f.: (1+x-2*x^2*y)/((1-x)^2*(1-x*y)^3). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 05 2004
%F A071943 G.f.=(1-q)/[z(2t+2t^2z-1+q)], where q=sqrt(1-4tz-4t^2z^2).
%F A071945 G.f.=(1-q)/[z(1+tz)(2t-1+q)], where q=sqrt(1-4tz-4t^2z^2).
%F A071947 G.f.=t(1+tz-q)/[(1+tz)(2t^2*z+tz-1+q)], where q=sqrt(1-2tz-3t^2*z^2).
%F A073149 G.f.: Sum_{n>=0,k>=0} T(n,k)*y^k*x^n = A(x)*A(xy)/(1-y) where A(x) is g.f. of A002212.
%F A078391 G.f.=C(z)C(tz), where C(z) = (1-sqrt(1-4z))/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004
%F A078436 G.f.: x*y*(2-x)/(1-2*x*y)/(1-x)^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 31 2002
%F A078803 G.f.: 1/[1-tz(1+z+z^2)]-1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004
%F A079213 G.f.: sum_{m>=0, n>=0, k>=0} f(m,n,k) x^m y^n z^k = (1+x)(1+y)/((1-x^2)(1-y^2)+x y z(1+x y)).
%F A079213 G.f. (conjectured): sum_{n>=0, k>=0} T(n,k) x^n y^k = sqrt((1+x)/((1+x-x y)((1-x)^2 - x y(1+x)))).
%F A080247 G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 16 2003
%F A081577 G.f.: 1/(1-x-y-2xy). - Ralf Stephan, Apr 28 2004
%F A082601 G.f.: x/(1-x-x^2*y-x^3*y^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 30 2003
%F A085472 G.f.: 1/2 Product(1+x^i*y^j), i,j>=0.
%F A085478 G.f.=(1-z)/[(1-z)^2-tz]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
%F A088326 G.f.: exp(sum_{k=1..infinity) z^k*B(x^k)/k ), where B(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + ... = G001190(x)/x - 1 and G001190 is the g.f. for the Wedderburn-Etherington numbers A001190.
%F A089052 G.f.: 1/Product_{k>=0} (1-y*x^(2^k)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 03 2003
%F A089353 G.f.: Prod(k=1..oo, 1/(1-q x^k)^k).
%F A089731 G.f.= g/(1+zg-tzg), where g := (1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4))/(2z^2) is the g.f. of A004148.
%F A089942 G.f.=(1+z-q)/[(1+z)(2z-t+tz+tq)], where q = sqrt(1-2z-3z^2).
%F A090842 G.f.: Sum_{k>=0} (1+x*y)/(1-x*y)/(1-(k+2)*x*y)*y^k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 12 2003
%F A090981 T(n,k)=binomial(n+1,k)*sum(binomial(n+1,j)*binomial(n-j-1,k-1),j=0..n-k)/(n+1). G.f. G=G(t,z) satisfies z(1-z+tz)G^2-(1-tz)G+1=0.
%F A090985 T(n,k)=binomial(n+k-2,k)*sum(binomial(n-2+k+i,i)*binomial(n-3-k-i,i-1), i=0..floor((n-2-k)/2))/(n-1). G.f. G=G(t,z) satisfies (1-t)G^3+(1+t)zG^2-z^2*(1+z)G+z^4=0.
%F A091187 T(n,k)=M(k-1)*binomial(n-1,k-1), where M(k)=A001006(k) = sum(binomial(k+1,q)*binomial(k+1-q,q-1),q=0..ceil((k+1)/2))/(k+1) is a Motzkin number. G.f. G=G(t,z) satisfies tzG^2-(1 - z + tz)G + 1- z + tz=0.
%F A091320 T(n,k)=(1/n)*binomial(n,k)*sum(2^(n+1-2k+j)*binomial(n,j)*binomial(n-k,k-1-j), j=0..n). G.f. G(t,z) satisfies zG^3 - (1 + z - tz)G + 1 = 0.
%F A091370 T(n,k)=[(k-1)/(n-k)]sum(2^j*binomial(n-2,n-k-1-j)*binomial(n-k,j),j=0..n-k-1). G.f.=t^3*z^3*S^2/(1-tzS), where S = [1+z-sqrt(1-6*z+z^2)]/(4z) is the g.f. of the little Schroder numbers (A001003).
%F A091435 G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 05 2004
%F A091438 G.f.: A(x,y) = prod{i=1..inf,j=1..i}(1/(1-x^i*y^j))
%F A091442 G.f.: A(x,y) = Product_{k>=1} (1-x^n)*(1-y^n)/(1-x^n-y^n).
%F A091467 G.f.: A(x,y) = Sum_{k>=1} (phi(k)/k)*log((1-x^n*y^n)^2/(1-x^n*y^n*(3+x^n+y^n))).
%F A091562 G.f. A(x,y) = (1-x-x*y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
%F A091836 G.f.=(1+zM)/[1-tz(1+zM)], where M=1+zM+z^2M^2 is the g.f. of the Motzkin numbers (A001006).
%F A091866 G.f. = G = G(t,z) satisfies z(1-tz)G^2-(1+z-2tz)G+1-tz = 0.
%F A091867 T(n,k)=[binomial(n+1,k)/(n+1)]*sum(binomial(n+1-k,j)*binomial(n-k-j-1,j-1), j=1..floor((n-k)/2)) for k<n; T(n,n)=1; T(n,k)=0 for k>n. G.f.=G=G(t,z) satisfies z(1+z-tz)G^2-(1+z-tz)G+1=0. T(n,k)=r(n-k)*binomial(n,k), where r(n)=A005043(n) are the Riordan numbers.
%F A091869 T(n,k)=binomial(n-1,k)*sum(binomial(n-k,j)*binomial(n-k-j,j-1),j=0..ceil((n-k)/2))/(n-k) for 0<=k<n; T(n,k)=0 for k>=n. G.f.=G=G(t,z) satisfies zG^2-(1+z-tz)G+1+z-tz=0. T(n,k)=M(n-k-1)*binomial(n-1,k), where M(n)=A001006(n) are the Motzkin numbers.
%F A091965 G.f.=G=2/[1-3z-2tz+sqrt(1-6z+5z^2)]. Alternatively, G=M/(1-tzM), where M=1+3zM+z^2*M^2.
%F A092276 T(n,k)=2k*binomial(3n-k,n-k)/(3n-k). G.f. = 1/(1-tzg^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.
%F A092392 G.f.: 2^k/[sqrt(1-4x)*(1+sqrt(1-4x))^k].
%F A092921 G.f.: x/[1-sum(i=0..k, x^i)].
%F A094021 T(n,k)=binomial(n,k-1)*binomial(3n-2k-1,n-k)/(2n-k). G.f. G=G(t,z) satisfies G^3+(t^3*z^2-t^2*z-3)G^2+(t^2*z+3)G-1=0.
%F A094112 T(n,k)=n!/[(k-2)!k] for 2<=k<=n-1; T(n,n)=n; T(n,1)=0 for n>=2; T(n,k)=0 for k>n. G.f. = sum(T(n,k)t^k z^n/n!, n,k>=1) = z[(t-1)exp(tz)+1]/(1-z).
%F A094322 G.f.=G=G(t,z)=(1-z)/(1-zC+z^2*C -tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%F A094368 G.f.: exp(z*arctan(x)) / sqrt(1+x^2).
%F A094449 G.f.=G=G(t,z)= (1-tz)(1-z)/[1-2tz+tz^2-z(1-z)(1-t*z)C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%F A096651 G.f.: A(x,y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], where P_n(y) is the n-th row polynomial of triangle A096800.
%F A096794 G.f. (1 - (1 - 4*x)^(1/2))/(3 - 2y + (2y-1)(1 - 4*x)^(1/2) ) = Sum_{n>=1,k>=0} a(n,k) x^n y^k.
%F A097094 T(n,0) = 1 and T(n,n) = A097095(n) for n>=0; T(n,k) = T(n-1,k) + T(n-2,k-1) for n>k>=1. G.f.: A(x,y) = A097095(x*y)/(1-x-x^2*y), where A097095(x)/(1-x-x^2) = A097095(x^2)^2/(1-x-x^3)^2, and A097095(x) is the g.f. of the main diagonal.
%F A097098 T(n,k)=(k+1)T(n-k,0) for k<n; T(n,n)=1. T(n,0)=a(n)-2a(n-1)+a(n-2) (n>=2), where a(n)=sum(binomial(k,n-k)*binomial(k,n-k+1)/k,k=ceil((n+1)/2)..n) = A004148(n). G.f.=1/(1-tz)+(1-z)(g-1-zg)/(1-tz)^2, where g=[1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. of the RNA secondary structure numbers (A004148).
%F A097179 G.f.: A(x,y) = 2*y/((1-8*x*y) + (2*y-1)*(1-8*x*y)^(3/4)). G.f.: A(x,y) = A004982(x*y)/(1 - x*A048779(x*y)).
%F A097181 G.f.: A(x,y) = 2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)). G.f.: A(x,y) = A097183(x*y)/(1 - x*A097184(x*y)).
%F A097186 G.f.: A(x,y) = 3*y/((1-9*x*y) + (3*y-1)*(1-9*x*y)^(2/3)). G.f.: A(x,y) = A004988(x*y)/(1 - x*A097188(x*y)).
%F A097190 G.f.: A(x,y) = 3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)). G.f.: A(x,y) = A097192(x*y)/(1 - x*A097193(x*y)).
%F A097609 G.f.=2/[1-2tz+z+sqrt(1-2z-3z^2)].
%F A097724 T(n,k)=(k+1)sum(binomial(j,n-k-j)*binomial(j+k,n+1-j)/j,j=ceil((n-k+1)/2)..n-k) for 0<=k<n; T(n,n)=1. G.f.=G/(1-tzG), where G = [1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. for the sequence A004148.
%F A097808 Columns have g.f. (1+2x)/(1+x)^2(x/(1+x))^k.
%F A098277 G.f.: Sum[n>=0, D(n,x)t^n] = 1/(1-2(x+1)t/(1-2(x+2)t/(1-4(x+3)t/(1-4(x+4)t/...)))).
%F A098358 G.f.: xy / [(1-x)^3 * (1-y)^3 ]. - Ralf Stephan, Oct 27 2004
%F A098359 G.f.: [xy(1+x)(1+y)] / [(1-x)^3 * (1-y)^3 ]. - Ralf Stephan, Oct 27 2004
%F A098360 G.f.: [xy(1+x+4x^2)(1+y+4y^2)] / [(1-x)^4 * (1-y)^4 ]. - Ralf Stephan, Oct 27 2004
%F A098432 G.f.: Sum[n>=0, S(n,x)t^n] = 1/(1+t-4*2(x+1)t/(1-4*2(x+2)t/(1+t-4*4(x+3)t/(1-4+4(x+4)t/...)))).
%F A098474 G.f.: 2/(1-x+(1-x-4*x*y)^(1/2)). E.g.f.: exp(x*(1+2*y))*(BesselI(0,2*x*y)-BesselI(1,2*x*y)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 11 2004
%F A099172 G.f.: (1 + xy + x^2y^2)/(1 - x - y + xy - x^2y^2).
%F A099509 G.f.: (1-x+x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2+x^3*y^2+x^4*y^4).
%F A099510 G.f.: (1-x+2*x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2-2*x^3*y^2+x^4*y^4). T(n,k) = binomial(2*n-2*(k\2),k).
%F A099512 G.f.: (1-x+3*x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2-7*x^3*y^2+x^4*y^4).
%F A099514 G.f.: (1-x+x*y-2*x^2*y^2)/((1-x)^2-4*x^2*y^2+3*x^3*y^2+4*x^4*y^4).
%F A099527 G.f.: (1-x*(2-3*y)-x^2*y^2)/(1-4*x+x^2*(4-2*y^2)-5*x^3*y^2+x^4*y^4).
%F A099557 G.f.: (1-x+x*y)/((1-x)^2-x^3*y^2).
%F A099602 G.f.: (1 + (y+1)*x - (y+1)*x^2)/(1 - (y+1)*(y+2)*x^2 + (y+1)^2*x^4).
%F A099605 G.f.: (1+2*(y+1)*x-(y+1)*x^2)/(1-(2*y+1)*(2*y+2)*x^2+(y+1)^2*x^4). T(n,n) = 2^n.
%F A100247 T(n,k) = A033184(n-[k/2],k) for n>0 (with A033184 formatted as a square array). G.f. A(x,y) satisfies: A(x^2,y)=((1+x)/(2*y-x*(1-sqrt(1-4*x*y)))-(1-x)/(2*y+x*(1-sqrt(1+4*x*y))))*y/x .
%F A059438 If g(x) = x+x^2+3*x^3+13*x^4+... is the generating function for the number of permutations with no global descents, then 1/(1-g(x)) is the generating function for n!. Setting t=1 in f(x,t) implies sum( T(n,k), k=1..n) = n!. Let g(x) be the o.g.f. for A003319. Then the o.g.f. for this table is given by f(x,t) = 1/(1-t*g(x))-1 (i.e. the coefficient of x^n*t^k in f(x,t) is T(n,k)). - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Jul 29 2004
%F A089962 T(n,k)=(-1)^(n-k)*C(n,k)*n*k^(n-k-1) for 0<k<=n, with T(0,0)=1. O.g.f.: A(x,y)=(1-y)*sum_{n>=0}x^n*y^n/(1+n*y)^(n+2).