A125275 Eigensequence of triangle A039599: a(n) = Sum_{k=0..n-1} A039599(n-1,k)*a(k) for n > 0 with a(0) = 1.

1, 1, 2, 7, 31, 162, 968, 6481, 47893, 386098, 3364562, 31460324, 313743665, 3320211313, 37124987124, 436985496790, 5397178181290, 69748452377058, 940762812167126, 13213888481979449, 192891251215160017

Offset

0,3

Comments

Starting with offset 1, these are the row sums of triangle A147294. - Gary W. Adamson, Nov 05 2008

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Formula

a(n) = Sum_{k=0..n-1} a(k) * C(2*n-1, n-k-1) * (2*k + 1)/(2*n - 1) for n > 0 with a(0) = 1.

Example

a(3) = 2*(1) + 3*(1) + 1*(2) = 7;
a(4) = 5*(1) + 9*(1) + 5*(2) + 1*(7) = 31;
a(5) = 14*(1) + 28*(1) + 20*(2) + 7*(7) + 1*(31) = 162.
Triangle A039599(n,k) = C(2*n+1, n-k)*(2*k+1)/(2*n+1) (with rows n >= 0 and columns k = 0..n) begins:
   1;
   1,  1;
   2,  3,  1;
   5,  9,  5,  1;
  14, 28, 20,  7, 1;
  42, 90, 75, 35, 9, 1;
  ...
where the g.f. of column k is G000108(x)^(2*k+1)
and G000108(x) = (1 - sqrt(1 - 4*x))/(2*x) is the Catalan g.f. function.

Mathematica

A125275=ConstantArray[0, 20]; A125275[[1]]=1; Do[A125275[[n]]=Binomial[2*n-1, n-1]/(2*n-1)+Sum[A125275[[k]]*Binomial[2*n-1, n-k-1]*(2*k+1)/(2*n-1), {k, 1, n-1}]; , {n, 2, 20}]; Flatten[{1, A125275}] (* Vaclav Kotesovec, Dec 09 2013 *)

Prog

(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, a(k)*binomial(2*n-1, n-k-1)*(2*k+1)/(2*n-1)))

Crossrefs

Cf. A000108, A039599, A125276 (variant), A147294.

Sequence in context: A221957 A030966 A009132 * A007446 A277396 A227119

Adjacent sequences: A125272 A125273 A125274 * A125276 A125277 A125278

Keyword

nonn

Author

Paul D. Hanna, Nov 26 2006

Status

approved