A251568 Expansion of e.g.f. exp(x*C(x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.

1, 1, 5, 43, 529, 8501, 169021, 4010455, 110676833, 3484717129, 123320412181, 4847038223171, 209536628422705, 9882471447634813, 505033804901100749, 27802319803528367791, 1640388588050579832001, 103275015543414629215505, 6910877628962983581031333

Offset: 0

Author

Paul D. Hanna, Dec 05 2014

Keywords

• nonn
• easy

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 529*x^4/4! + 8501*x^5/5! + ...
where
log(A(x)) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ... + A000108(n)*x^n + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..360

Crossrefs

Cf. A080893, A250917.

Cf. A082579, A380511, A380514.

Cf. A001517, A251569, A000108, A382032.

Programs

Maple

CatalanNumber := n -> binomial(2*n,n)/(n+1):
a := n -> `if`(n=0, 1, n!*CatalanNumber(n)*hypergeom([1-n], [2+n], -1)):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 04 2017

Mathematica

Flatten[{1,Table[Sum[n!/k!*Binomial[2*n-1,n-k]*2*k/(n+k),{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 14 2015 *)
a[0] = 1; a[n_] := (2n)!/(n+1)! Hypergeometric1F1[1-n, n+2, -1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 03 2017, after Vladimir Kruchinin *)

PARI

{a(n)=my(C=1);for(i=1,n,C=1+x*C^2 +x*O(x^n));n!*polcoef(exp(x*C^2),n)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n) = if(n==0, 1, sum(k=1, n, n!/k! * binomial(2*n-1, n-k) * 2*k/(n+k) ))}
for(n=0, 20, print1(a(n), ", "))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(serreverse(x*(1-x))^2/x))) \\ Seiichi Manyama, Mar 15 2025

Formula

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

E.g.f. A(x) satisfies: A'(x)/A(x) = C'(x) = C(x)^2 / sqrt(1-4*x) where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function.

Recurrence equation: a(n) = -(n^2 - 5*n +1)*a(n-1) + n*(2*n - 3)*(2*n - 4)*a(n-2) with a(0) = 1, a(1) = 1. It appears that a(n) - 1 is divisible by n*(n - 1) for n >= 2. - Peter Bala, Feb 14 2015

a(n) ~ 2^(2*n+1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Feb 14 2015

a(n) are special values of the hypergeometric function 1F1: a(n) = 4^n*Gamma(n+1/2)*exp(-1)*hypergeom([2*n+1], [n+2], 1)/(sqrt(Pi)*(n+1)), for n>=1. - Karol A. Penson, Jun 01 2015

a(n) = ((2*n)!/(n+1)!)*hypergeometric([1-n],[n+2],-1), a(0)=1. - Vladimir Kruchinin, May 03 2017

From Seiichi Manyama, Mar 15 2025: (Start)

E.g.f.: exp( (1/x) * Series_Reversion( x*(1-x) )^2 ).

E.g.f.: exp( Series_Reversion( x/(1+x)^2 ) ). (End)