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A138736 Inverse binomial transform of A138737. 2


%S 1,1,4,36,368,5200,90432,1884736,45817088,1273874688,39891461120,

%T 1389816423424,53334303584256,2235679577657344,101651458028158976,

%U 4983219643056537600,262026143585449607168,14711289584591513387008

%N Inverse binomial transform of A138737.

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

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

%H Vaclav Kotesovec, <a href="/A138736/b138736.txt">Table of n, a(n) for n = 0..300</a>

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

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

%F a(n) ~ (1 + LambertW(exp(-1)))^(3/2)*n^(n-1) / (exp(n-2)*LambertW(exp(-1))^(n-1)). - _Vaclav Kotesovec_, Oct 30 2017

%o (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]}

%Y Cf. A138737.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 05 2008

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Last modified May 27 13:00 EDT 2020. Contains 334657 sequences. (Running on oeis4.)