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%I #21 Nov 16 2017 01:43:25
%S 1,0,4,0,176,0,24192,0,6966528,0,3459768320,0,2642804895744,0,
%T 2880385956446208,0,4248075487709560832,0,8150896954121845211136,0,
%U 19738484719911211593891840,0,58891248341123034677673721856,0
%N a(n) = n-th term of the (n+1)-th inverse binomial transform of A138737 for n>=0.
%C The n-th term of the n-th inverse binomial transform of A138737 = (n+1)^(n-1) for n>=0.
%H Vincenzo Librandi, <a href="/A138734/b138734.txt">Table of n, a(n) for n = 0..200</a>
%F From _Paul D. Hanna_, Apr 20 2011: (Start)
%F a(n) = Sum_{k=0..n} (-1)^k*C(n,k) * (k+1)^(k-1) * (n-k+1)^(n-k-1).
%F E.g.f.: -LambertW(x)*LambertW(-x)/x^2.
%F E.g.f.: exp( Sum_{n>=1} 2*(2n)^(2n-1) * x^(2n)/(2n)! ). (End)
%F Let Chw(x) = Sum_{n>=0} (2*n+1)^(2*n-1) * x^(2*n)/(2*n)!
%F and Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!
%F then e.g.f. A(x) = Chw(x)^2 - Shw(x)^2. - _Paul D. Hanna_, Aug 22 2012
%F If n is even, a(n) ~ 2*exp(2) * LambertW(exp(-1)) * n^(n-1). - _Vaclav Kotesovec_, Oct 08 2013
%e E.g.f.: A(x) = 1 + 4*x^2/2! + 176*x^4/4! + 24192*x^6/6! + 6966528*x^8/8! +...
%t CoefficientList[Series[-LambertW[x]*LambertW[-x]/x^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 08 2013 *)
%o (PARI) {a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=(k+1)^(k-1)-polcoeff(subst(Ser(A),x,x/(1+k*x+x*O(x^k)))/(1+k*x),k)); polcoeff(subst(Ser(A),x,x/(1+(n+1)*x+x*O(x^n)))/(1+(n+1)*x),n)}
%o (PARI) {a(n)=sum(k=0, n, (-1)^k*binomial(n,k)*(k+1)^(k-1)*(n-k+1)^(n-k-1))}
%o (PARI) {a(n)=local(LW=sum(m=1,n+1,m^(m-1)*x^m/m!)+x^2*O(x^n));n!*polcoeff(-LW*subst(LW,x,-x)/x^2,n)}
%o (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n,2*(2*m)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n)),n)}
%o (PARI) x='x+O('x^50); Vec(serlaplace(-lambertw(x)*lambertw(-x)/x^2)) \\ _G. C. Greubel_, Nov 15 2017
%Y Cf. A138737, A215880.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 05 2008
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