A321089 G.f.: Sum_{n>=0} ((1+x)^(n+1) - 1)^n.

1, 2, 10, 82, 928, 13406, 235690, 4883702, 116548222, 3148151702, 94950591878, 3162966582742, 115334767261792, 4569294561813770, 195438629679894238, 8975996556375735458, 440572146080811981406, 23015418712779922737206, 1274980039012724226987966, 74655326188457739033712062, 4607114081638141934903219532, 298862442692043953057588327202

Offset: 0

Author

Paul D. Hanna, Nov 04 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 82*x^3 + 928*x^4 + 13406*x^5 + 235690*x^6 + 4883702*x^7 + 116548222*x^8 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..375

Programs

PARI

{a(n) = polcoeff( sum(k=0, n, ((1+x +x*O(x^n))^(k+1) - 1)^k), n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* From e.g.f. infinite series: */
\p200 \\ set precision
{A = Vec(round( sum(n=0, 600, 1./(1 + (1+x +O(x^26))^(-n))^(n+1)) ))}
for(n=0, #A-1, print1(A[n+1], ", "))

Formula

Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n)^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654288138301722... and c = 0.8785394171057422507960514834733179025314463... - Vaclav Kotesovec, Oct 04 2020