A001002

[Emeric Deutsch]

Dear Seqfans,
Rather ashamed to ask but library is far and visit need not be
fruitful.
What are these dissections?
For example, how do we get the 3 and the 10?
Thanks a lot.
Emeric


ID Number: A001002 (Formerly M2852 and N1146)
URL:       http://www.research.att.com/projects/OEIS?Anum=A001002
Sequence:  1,1,3,10,38,154,654,2871,12925,59345,276835,1308320,6250832,
            30142360,146510216,717061938,3530808798,17478955570,
            86941210950,434299921440,2177832612120,10959042823020,
            55322023332420,280080119609550
Name:      Number of dissections of a polygon.
Comments:  G.f. (offset 1) is series reversion of x-x^2-x^3.
            Antidiagonal sums of triangle A104978 which has g.f. F(x,y) 
that
               satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D Hanna
               (pauldhanna(AT)juno.com), Mar 30 2005
References I. M. H. Etherington, Some problems of non-associative 
combinations,
               Edinburgh Math. Notes, 32 (1940), 1-6.
            T. S. Motzkin, Relations between hypersurface cross ratios, and 
a
               combinatorial formula for par titions of a polygon, for 
permanent
               preponderance, and for non-associative products, Bull. Amer. 
Math.
               Soc., 54 (1948), 352-360.
Links:     INRIA Algorithms Project, Encyclopedia of Combinatorial 
Structures 395
            Index entries for reversions of series
Formula:   a(n)=sum(binomial(n+k,k)*binomial(k,n-k),k=ceil(n/2)..n)/(n+1);
               5n(n+1) * a(n) = 11n(2n-1) * a(n-1) + 3(3n-2)(3n-4) * a(n-2) 
- Len Smiley
               (smiley(AT)math.uaa.alaska.edu)
            G.f.: (4sin(asin((27x+11)/16)/3)-1)/(3x); - Paul Barry
               (pbarry(AT)wit.ie), Feb 02 2005
            a(n) = Sum_{k=0..[n/2]} C(2*n-k,n+k)*C(n+k,k)/(n+1). - Paul D 
Hanna
               (pauldhanna(AT)juno.com), Mar 30 2005
Math'ca:   Rest[CoefficientList[InverseSeries[Series[y - y^2 - y^3, {y, 0,
               30}], x], x]]
Program:   (PARI)

a(n)=if(n<0,0,polcoeff(serreverse(x-x^2-x^3+x^2*O(x^n)),n+1))
            (PARI)
               a(n)=if(n<0,0,sum(k=0,n\2,(2*n-k)!/k!/(n-2*k)!)/(n+1)!)
            (PARI)
               a(n)=sum(k=0,n\2,binomial(2*n-k,n+k)*binomial(n+k,k))/(n+1)
               (Hanna)
See also:  n*a(n)=A038112(n-1), n>0.
            Cf. A104978.
            Adjacent sequences: A000999 A001000 A001001 this_sequence 
A001003
               A001004 A001005
            Sequence in context: A092816 A078109 A083692 this_sequence 
A000902
               A103138 A074527
Keywords:  nonn,easy,nice
Offset:    0
Author(s): njas