A191460 E.g.f.: A(x) = Sum_{n>=0} binomial(x^n, n).

1, 1, -1, 2, 6, 24, -480, 720, 13440, 100800, -1874880, 3628800, 22619520, 479001600, -36739422720, 468583315200, 6345334195200, 20922789888000, -4289844445286400, 6402373705728000, 363873320191180800, 18895538930171904000, -350378437544239104000

Offset

0,4

Comments

The definition of the e.g.f. A(x) is an application of the identity:

* Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} binomial(q^n, n)*x^n at x=1, q=x.

Links

Table of n, a(n) for n=0..22.

Formula

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

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

|a(p)| = (p-1)! for prime p.

Example

E.g.f.: A(x) = 1 + x - x^2/2! + 2*x^3/3! + 6*x^4/4! + 24*x^5/5! +...
Series expansions:
* A(x) = 1 + log(1 + x) + log(1 + x^2)^2/2! + log(1 + x^3)^3/3! + log(1 + x^4)^4/4! +...+ log(1 + x^n)^n/n! +...
* A(x) = 1 + x + x^2*(x^2-1)/2! + x^3*(x^3-1)*(x^3-2)/3! + x^4*(x^4-1)*(x^4-2)*(x^4-3)/4! +...+ binomial(x^n, n) +...
Coefficients a(n)/n! in the series expansion of the e.g.f. begin:
A(x) = 1 + x - 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/5*x^5 - 2/3*x^6 + 1/7*x^7 + 1/3*x^8 + 5/18*x^9 - 31/60*x^10 + 1/11*x^11 + 17/360*x^12 + 1/13*x^13 +...
which illustrates the property |a(p)/p!| = 1/p for prime p.

Prog

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

Crossrefs

Cf. A191461.

Sequence in context: A274098 A324141 A277423 * A073475 A061774 A110729

Adjacent sequences: A191457 A191458 A191459 * A191461 A191462 A191463

Keyword

sign

Author

Paul D. Hanna, Jun 02 2011

Status

approved