A202360 E.g.f.: A(x) = Sum_{n>=0} 2^n * binomial(x^n + n-1, n).

1, 2, 4, 16, 144, 768, 10560, 92160, 1585920, 21127680, 419973120, 7431782400, 177081569280, 3923981107200, 105929096232960, 2868863206809600, 87449689674547200, 2742391916199936000, 94359281224797388800, 3356687705428721664000

Offset

0,2

Comments

The definition of the e.g.f. A(x) is an application of the identity Sum_{n>=0} (-1)^n*log(1 - q^n*x)^n*y^n/n! = Sum_{n>=0} binomial(q^n*y +n-1, n)*x^n at x=2, y=1, q=x.

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: A(x) = Sum_{n>=0} (-1)^n * log(1 - 2*x^n)^n/n!.

E.g.f.: A(x) = Sum_{n>=0} 2^n * Sum_{k=0..n} |Stirling1(n,k)|*x^(n*k)/n!.

Example

 E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 16*x^3/3! + 144*x^4/4! + 768*x^5/5! +...
Series expansions:
A(x) = 1 - log(1 - 2*x) + log(1 - 2*x^2)^2/2! - log(1 - 2*x^3)^3/3! + log(1 - 2*x^4)^4/4! +...+ (-1)^n*log(1 - 2*x^n)^n/n! +...
A(x) = 1 + 2*x + 2^2*x^2*(x^2+1)/2! + 2^3*x^3*(x^3+1)*(x^3+2)/3! + 2^4*x^4*(x^4+1)*(x^4+2)*(x^4+3)/4! +...+ 2^n*binomial(x^n + n-1, n) +...

Prog

 (PARI) {a(n)=local(A=1+x); A=sum(m=0, n, 2^n*binomial(x^m+n-1 +x*O(x^n), m)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); A=1+sum(m=1, n, (-1)^m*log(1-2*x^m +x*O(x^n))^m/m!); n!*polcoeff(A, n)}
(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, 2^m*sum(k=0, m, abs(Stirling1(m, k))*x^(m*k))/m!)); n!*polcoeff(A, n)}

Crossrefs

Cf. A191461.

Sequence in context: A073924 A362065 A061588 * A050472 A109457 A352801

Adjacent sequences: A202357 A202358 A202359 * A202361 A202362 A202363

Keyword

nonn

Author

Paul D. Hanna, Dec 17 2011

Status

approved