A138736 Inverse binomial transform of A138737.

1, 1, 4, 36, 368, 5200, 90432, 1884736, 45817088, 1273874688, 39891461120, 1389816423424, 53334303584256, 2235679577657344, 101651458028158976, 4983219643056537600, 262026143585449607168, 14711289584591513387008

Offset: 0

Author

Paul D. Hanna, Apr 05 2008

Keywords

• nonn

Comments

The n-th term of the n-th inverse binomial transform of A138737 equals (n+1)^(n-1) for n>=0.

Related to LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A138737.

Programs

PARI

{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=(k+1)^(k-1)-Vec(subst(Ser(A),x,x/(1+(k-1)*x+x*O(x^k)))/(1+(k-1)*x))[k+1]);A[n+1]}

Formula

O.g.f. satisfies: [x^n] A( x/(1+(n-1)*x) )/(1+(n-1)*x) = (n+1)^(n-1) for n>=0.

E.g.f. satisfies: [x^n] A(x)*exp(-(n-1)*x) = (n+1)^(n-1)/n! for n>=0.

a(n) ~ (1 + LambertW(exp(-1)))^(3/2)*n^(n-1) / (exp(n-2)*LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Oct 30 2017