# q. about g.f.'s from a correspondent

Ralf Stephan ralf at ark.in-berlin.de
Sat Mar 3 12:22:15 CET 2007

```Just grep for g.f. and sqrt in the raw database:

some false positives are easily removed.

eis/eisBTfry00001.txt:%F A111940 The g.f. of column k of matrix power P^m (ignoring leading zeros) is: cos(m*acos(1-x^2/2))+(-1)^k*sin(m*acos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4).
eis/eisBTfry00002.txt:%F A108044 T(n, k)=binomial(n, (n+k)/2) if n+k is even, T(n, k)=0 if n+k is odd. G.f.=f/(1-tg), where f=1/sqrt(1-4x^2) and g=(1-sqrt(1-4x^2))/(2x). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 05 2005
eis/eisBTfry00002.txt:%F A025247 G.f.: (1+2*x-sqrt(1-4*x+4*x^2-4*x^3))/2 - Michael Somos, Jun 08, 2000.
eis/eisBTfry00003.txt:%F A025251 G.f.: (1+x^2-sqrt(1-2*x^2-8*x^3+x^4))/2 - Michael Somos, Jun 08, 2000.
eis/eisBTfry00003.txt:%C A100225 More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2.
eis/eisBTfry00003.txt:%F A100225 G.f.: (1+2*x+sqrt(1+8*x^2))/2. G.f.: A(x) = x/(series_reversion[x*(1-x)/(1-2*x-x^2)]). a(n) = -8*(n-3)*a(n-2)/n for n>2, with a(0)=1, a(1)=1, a(2)=2. a(2*n) = 2^n*(-1)^(n-1)*A000108(n-1), a(2*n+1)=0, for n>=1, where A000108=Catalan.
eis/eisBTfry00003.txt:%F A105523 G.f.: (1+2x-sqrt(1+4x^2))/(2x)
eis/eisBTfry00003.txt:%F A104506 G.f.: ( (1-x)/sqrt(1-2*x+5*x^2) - 1)/(2*x). a(n) = (-1)^n*n*A007440(n) (reversion of g.f. for Fibonacci numbers).
eis/eisBTfry00003.txt:%C A088138 The sequence 0,1,-2,0,8,-16,... has G.f. 1/(1+2x-4x^2), a(n)=2^n*sin(2n*pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)x)/sqrt(3):0,1,-3,0,9,...
eis/eisBTfry00003.txt:%F A088138 G.f.: 1/(1-2x+4x^2); E.g.f. : exp(x)sin(sqrt(3)x)/sqrt(3); a(n)=2a(n-1)-4a(n-2), a(0)=0, a(1)=1; a(n)=((1+isqrt(3))^n-(1-isqrt(3))^n)/(2isqrt(3)); a(n)=Im{(1+isqrt(3))^n/sqrt(3)}; a(n)=sum{k=0..floor(n/2), C(n, 2k+1)(-3)^k}.
eis/eisBTfry00004.txt:%F A080934 G.f. of n-th row: B(n)/B(n+1) where B(j)=[(1+sqrt(1-4x))/2]^j-[(1-sqrt(1-4x))/2]^j.
eis/eisBTfry00004.txt:%C A096794 Column k has g.f . F(x)^(k+1)*(2y)^k where F(x)=(1-sqrt(1-4*x))/(3-sqrt(1-4*x)) is the g.f. for Fine's sequence A000957.
eis/eisBTfry00004.txt:%C A111595 The row polynomials (1/2^n)* H(n,sqrt(x/2))^2, with the Hermite polynomials H(n,x), have e.g.f. (1/sqrt(1-y^2))*exp(x*y/(1+y)).
eis/eisBTfry00004.txt:%F A110503 G.f. for column k of matrix power A110503^m (ignoring leading zeros): cos(m*acos(1-x^2/2)) + (-1)^k*sin(m*acos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4)*(1+x)/(1-x).
eis/eisBTfry00005.txt:%F A112707 G.f. for column m>=0 (without leading zeros): c(-m*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
eis/eisBTfry00005.txt:%F A025259 G.f.: (1+2x-x^2-sqrt(1-4x+6x^2-8x^3+x^4))/(2x)=2-x+x^2*c(x^2/(1-x)^4)/(1-x)^2, c(x) the g.f. of A000108; a(n+3)=sum{k=0..floor(n/2), C(n+3k+1,6k+1)C(k)}, where C(n)=A000108(n). - Paul Barry (pbarry(AT)wit.ie), May 31 2006
eis/eisBTfry00005.txt:%F A025242 G.f.: (1+2*x+x^2-sqrt(1-4*x+2*x^2+x^4))/2 - Michael Somos, Jun 08, 2000.
eis/eisBTfry00005.txt:%F A104559 G.f.: A(x, y) = 2/(1-x+x^2*y^2 - 2*x*y + sqrt((1-x+x^2*y^2)^2 - 4*x^2*y^2)) (due to Emeric Deutsch). T(n, k) = C(n-[k/2], [(k+1)/2])*C(n-[(k+1)/2], [k/2]) = A104557(n, k)/(n-k)!.
eis/eisBTfry00005.txt:%F A025243 G.f.: (1+x+2*x^2-sqrt(1-2*x-3*x^2+4*x^4))/2 - Michael Somos, Jun 08, 2000.
eis/eisBTfry00006.txt:%F A105632 G.f.: A(x, y) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x*y)))/(2*x^2).
eis/eisBTfry00006.txt:%F A091491 G.f.: 2/(2-y*(1-sqrt(1-4*x)))/(1-x). T(n, k) = T(n-1, k-1) + T(n, k+1) for n>0, with T(0, 0)=1.
eis/eisBTfry00006.txt:%F A112705 G.f. for column m>=0 (without leading zeros): c(m*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
eis/eisBTfry00006.txt:%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)))).
eis/eisBTfry00006.txt:%F A025264 G.f.: (1-sqrt(1-8*x+12*x^2+12*x^3))/2 - Michael Somos, Jun 08, 2000.
eis/eisBTfry00006.txt:%F A094876 a(6n-1) = a(6n+1) = a(6n+2) = a(6n+3) = a(6n+4) = a(6n+6) = A000108(n) except a(-1) = 0; generating function = (1-sqrt(1-4x^6))(1+x^2+x^3+x^4+x^5+x^7)/(2x^7)-1/x. - Alec Mihailovs (alec(AT)mihailovs.com), Jun 16 2004
eis/eisBTfry00007.txt:%F A089408 G.f.: (1+4x-(1+2x)sqrt(1-4x^2))/(2x) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2005
eis/eisBTfry00007.txt:%F A063894 G.f. A(x)=1-sqrt(1-4x+A(x^2)) satisfies A(x)^2-2A(x)+4x-A(x^2)=0, A(0)=0. - Michael Somos, Sep 06 2003
eis/eisBTfry00007.txt:%F A115141 O.g.f.: 1/c(x)^2 = (1-x) - x*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers).
eis/eisBTfry00007.txt:%C A115139 These polynomials appear in the formula 1/c(x)^n = P(n+1,x) - x*P(n,x)*c(x), n>=1, with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers). See the W. Lang reference, eqs. (1) and (2), p. 408, with P(n,x):=p(-n,x).
eis/eisBTfry00007.txt:%C A111963 Row sums have g.f. 1/sqrt(1+4x^2) [alternating sign central binomial numbers with interpolated zeros]. Diagonal sums are A111964. Inverse of A111960. Factors as (1/sqrt(1+4x^2),x/sqrt(1+4x^2))*(1/(1+x),x/(1+x)).
eis/eisBTfry00008.txt:%C A100229 The main diagonal forms A100230. Secondary diagonal is T(n+1,n) = (n+1)*A052924(n). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z, and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).
eis/eisBTfry00008.txt:%F A025228 G.f.: (1-sqrt(1-8*x+12*x^2))/2 - Michael Somos, Jun 08, 2000.
eis/eisBTfry00008.txt:%C A104505 Matrix inverse is triangle A104509 and is related to Fibonacci numbers. Column 0 equals A098331, with g.f.: 1/sqrt(1-2*x+5*x^2). Column 1 equals A104506, with g.f.: ((1-x)/sqrt(1-2*x+5*x^2)-1)/(2*x). Row sums equal A104507. Absolute row sums equal A104508.
eis/eisBTfry00008.txt:%C A104505 Array (1/sqrt(1-2x+5x^2), (1-x-sqrt(1-2x+5x^2))/(2x)), in Riordan array notation. Product of A120616 by A007318. Column k has e.g.f. exp(x)Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k. - Paul Barry (pbarry(AT)wit.ie), Jun 17 2006

and so on, you get the idea...

ralf

Thanks to everyone for these comments and corrections!

I will revise that entry

Neil

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