A125276 Eigensequence of triangle A039598: a(n) = Sum_{k=0..n-1} A039598(n-1,k)*a(k) for n>0 with a(0)=1.

1, 1, 3, 12, 58, 325, 2060, 14514, 112170, 941128, 8502393, 82160481, 844532873, 9191329357, 105491177081, 1272418794619, 16080824798705, 212370154398094, 2923859710010527, 41877072960374478, 622763691600244335

Offset: 0

Author

Paul D. Hanna, Nov 26 2006

Keywords

• nonn

Examples

			a(3) = 5*(1) + 4*(1) + 1*(3) = 12;
a(4) = 14*(1) + 14*(1) + 6*(3) + 1*(12) = 58;
a(5) = 42*(1) + 48*(1) + 27*(3) + 8*(12) + 1*(58) = 325.
Triangle A039598(n,k) = C(2*n+2,n-k)*(k+1)/(n+1) begins:
1;
2, 1;
5, 4, 1;
14, 14, 6, 1;
42, 48, 27, 8, 1;
132, 165, 110, 44, 10, 1; ...
where g.f. of column k = G000108(x)^(2*k+2)
and G000108(x) = (1 - sqrt(1-4*x))/(2x) is the Catalan function.
		

Links

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

Crossrefs

Cf. A039598, A000108; A125275 (variant).

Programs

Mathematica

A125276=ConstantArray[0,20]; A125276[[1]]=1; Do[A125276[[n]]=Binomial[2*n,n-1]/n+Sum[A125276[[k]]*Binomial[2*n,n-k-1]*(k+1)/n,{k,1,n-1}];,{n,2,20}]; Flatten[{1,A125276}] (* Vaclav Kotesovec, Dec 09 2013 *)

PARI

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

Formula

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

G.f. A(x) satisfies: A(x/(1+x)^2) = 1 + x*A(x); also, A(x*(1-x)) = 1 + [x/(1-x)]*A(x/(1-x)); also, A(x) = 1 + x*C(x)^2*A(x*C(x)^2) where C(x) = (1 - sqrt(1-4x))/(2x) is the Catalan function (A000108). - Paul D. Hanna, Aug 15 2007