A118794 E.g.f.: 1 - exp((-1 + sqrt(5 - 4*exp(x)) )/2).

0, 1, 2, 11, 121, 1902, 38381, 945989, 27552260, 925920081, 35265751869, 1501219998148, 70632987480771, 3639861179067661, 203881981765871618, 12333901891547136559, 801418950244634922973, 55665376886060309513990

Offset: 0

Author

Paul D. Hanna, Apr 30 2006

Keywords

• nonn

Comments

Also equals the row sums of triangle A118793 (offset without leading zero).

Examples

			E.g.f.: A(x) = x + 2/2*x^2 + 11/6*x^3 + 121/24*x^4 + 1902/120*x^5 + ...
		

Crossrefs

Cf. A118793, A118795.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[1-Exp[(Sqrt[5-4Exp[x]]-1)/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 07 2014 *)

Maxima

a(n):=sum(((-1)^(m+1)*sum(((k+m-1)!*binomial(2*k+m-1,k+m-1)*stirling2(n,k+m)),k,0,n-m))/(m-1)!,m,1,n); /* Vladimir Kruchinin, Jul 02 2011 */

PARI

{a(n)=local(x=X+X^2*O(X^n));n!*polcoeff(1-exp((-1+sqrt(5-4*exp(x)))/2),n,X)}
for(n=0,30,print1(a(n),", "))

PARI

/* As the row sums of A118793: */
{a(n)=local(x=X+X^2*O(X^n));if(n<1,0, -(n-1)!*sum(k=0,n-1,polcoeff(((x/log(1-x-x^2)))^n/(n-1-k)!,k,X)))}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = (n-1)!*Sum_{k=0..n-1} [x^k] (x/log(1-x-x^2))^n/(n-1-k)! for n>0.

a(n) = sum(m=1..n, ((-1)^(m+1)*sum(k=0..n-m, ((k+m-1)!*binomial(2*k+m-1,k+m-1)*stirling2(n,k+m))))/(m-1)!). - Vladimir Kruchinin, Jul 02 2011

a(n) ~ sqrt(5/2) * n^(n-1) / (2 * exp(n+1/2) * (log(5/4))^(n-1/2)). - Vaclav Kotesovec, Jul 31 2014