OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..500
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
EXAMPLE
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.
MATHEMATICA
PROG
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, a(k)*binomial(2*n, n-k-1)*(k+1)/n))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 26 2006
STATUS
approved