A191461 E.g.f.: A(x) = Sum_{n>=0} binomial(x^n + n-1, n).

1, 1, 1, 2, 18, 24, 480, 720, 23520, 100800, 1874880, 3628800, 341953920, 479001600, 36739422720, 468583315200, 8960682931200, 20922789888000, 4289844445286400, 6402373705728000, 1012647189038284800, 18895538930171904000, 350378437544239104000

Offset

0,4

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=1, y=1, q=x.

Links

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

Formula

E.g.f.: A(x) = Sum_{n>=0} (-1)^n * 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! + 18*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! +...+ (-1)^n*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-1, n) +...
Coefficients a(n)/n! in the series expansion of the e.g.f. begin:
A(x) = 1 + 1/2*x + 1/3*x^2 + 3/4*x^3 + 1/5*x^4 + 2/3*x^5 + 1/7*x^6 + 7/12*x^7 + 5/18*x^8 + 31/60*x^9 + 1/11*x^10 + 257/360*x^11 + 1/13*x^12 +...
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, 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-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, sum(k=0, m, abs(Stirling1(m, k))*x^(m*k))/m!)); n!*polcoeff(A, n)}

Crossrefs

Cf. A191460.

Sequence in context: A063430 A031104 A115042 * A366957 A098561 A141426

Adjacent sequences: A191458 A191459 A191460 * A191462 A191463 A191464

Keyword

nonn

Author

Paul D. Hanna, Jun 02 2011

Status

approved