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%I #25 Oct 09 2017 17:12:01
%S 1,6,52,600,8656,149856,3026752,69866880,1814338816,52350752256,
%T 1661575754752,57531530434560,2158011794968576,87173881613869056,
%U 3772959800981143552,174183372619165040640,8543978588021450407936,443748799382401230176256
%N Binomial convolution of the numbers in sequence A080253.
%H Vincenzo Librandi, <a href="/A217486/b217486.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = sum(binomial(n,k)*c(k)*c(n.k),k=0..n), where c(n) = A080253(n).
%F a(n) = 2^n*t(n+1), where t(n) = ordered Bell numbers (A000670).
%F E.g.f. exp(2*x)/(2-exp(2*x))^2.
%F G.f.: 1/G(0) where G(k) = 1 - x*3*(2*k+2) + x^2*(k+1)*(k+2)*(1-3^2)/G(k+1) ; (continued fraction due to T. J. Stieltjes). - _Sergei N. Gladkovskii_, Jan 11 2013.
%F a(n) ~ n!*n*2^(n-1)/(log(2))^(n+2). - _Vaclav Kotesovec_, Aug 11 2013
%t t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[Binomial[n,k]c[k]c[n-k], {k,0,n}], {n,0,100}]; Table[2^n t[n+1], {n,0,100}]
%t With[{nn=20},CoefficientList[Series[Exp[2x]/(2-Exp[2x])^2,{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Oct 09 2017 *)
%o (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
%o makelist(sum(binomial(n,k)*c(k)*c(n-k),k,0,n),n,0,10);
%o makelist(2^n*t(n+1),n,0,40);
%o (Sage)
%o def A217486(n):
%o return 2^n*add(add((-1)^(j-i)*binomial(j,i)*i^(n+1) for i in range(n+2)) for j in range(n+2))
%o [A217486(n) for n in range(18)] # _Peter Luschny_, Jul 22 2014
%Y Cf. A080253, A000670, A217483, A217484, A217485, A217487, A217488.
%K nonn
%O 0,2
%A _Emanuele Munarini_, Oct 04 2012
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