A232549 E.g.f.: Sum_{n>=0} ( -log(1 - n*x)/n )^n / n!.

1, 1, 2, 9, 69, 780, 12015, 240254, 6022121, 184115728, 6718994385, 287695076790, 14253791802085, 807709767584928, 51832973461142601, 3734821283563097350, 299928689437970776785, 26669987585007924913536, 2610872906937221730434977, 279949862628226582759321542, 32727135115533487802060517885

Offset: 0

Author

Paul D. Hanna, Nov 25 2013

Keywords

• nonn

Comments

Compare e.g.f. to the identity:

if G(x) = Sum_{n>=0} Integral( G(x)^t dx )^n/n!, then G(x)^t = 1/(1 - t*x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 69*x^4/4! + 780*x^5/5! + 12015*x^6/6! +...
where
A(x) = 1 - log(1-x) + log(1-2*x)^2/(2!*2^2) - log(1-3*x)^3/(3!*3^3) + log(1-4*x)^4/(4!*4^4) - log(1-5*x)^5/(5!*5^5) + log(1-6*x)^6/(6!*6^6) +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A128943.

Programs

Mathematica

Flatten[{1,Table[Sum[(-1)^(n-k) * k^(n-k) * StirlingS1[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 23 2014 *)

PARI

{a(n)=local(A=1+x);A=sum(m=0,n,intformal( 1/(1-m*x + x*O(x^n)))^m/m!);n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

PARI

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0, n, (-1)^(n-k) * k^(n-k) * Stirling1(n, k))}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} Integral( 1/(1 - n*x) dx )^n/n!, where the constant of integration is zero.

a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(n-k) * Stirling1(n,k).