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%I #19 Feb 06 2021 16:35:14
%S 0,1,12,117,1168,12525,145836,1844017,25238592,372320793,5894109100,
%T 99712850061,1795703361552,34303011804997,692863434782988,
%U 14754105717057225,330351159979499776,7758672154410196017,190717243734190845132,4896738903385469500453
%N a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^3.
%C Let b(n) denote the cubes (A000578). If e.g.f. of b(n) is E(x) and a(n) = Sum{k=0..n} C(n,k)^2*(n-k)!*b(k), then e.g.f. of a(n) is E(x/(1-x))/(1-x). (Thanks to _Vladeta Jovovic_ for help.) - _Miklos Kristof_, Apr 19 2005
%H Seiichi Manyama, <a href="/A105218/b105218.txt">Table of n, a(n) for n = 0..442</a>
%F E.g.f. = (x/(1-x)^2+3*x^2/(1-x)^3+x^3/(1-x)^4)*exp(x/(1-x)) - _Miklos Kristof_, Apr 19 2005
%F Recurrence: (n-2)*(n-1)^2*a(n) = (n-2)*n^2*(2*n-1)*a(n-1) - (n-1)^3*n^2*a(n-2). - _Vaclav Kotesovec_, Sep 26 2013
%F a(n) ~ n^(n+7/4)*exp(2*sqrt(n)-n-1/2)/sqrt(2) * (1 - 5/(48*sqrt(n))). - _Vaclav Kotesovec_, Sep 26 2013
%F a(n) = n*n!*hypergeom([2, 1-n], [1, 1], -1). - _Peter Luschny_, Apr 01 2015
%e b(n) = 0, 1, 8, 27, 64, 125, 216, ...
%e a(3) = C(3,0)^2*3!*b(0) + C(3,1)^2*2!*b(1) + C(3,2)^2*1!*b(2) + C(3,3)^2*0!*b(3) = 1*6*0 + 9*2*1 + 9*1*8 + 1*1*27 = 0 + 18 + 72 + 27 = 117.
%p seq(add(binomial(n,k)^2*(n-k)!*k^3, k=0..n),n=0..30);
%p # Alternatively:
%p a := n -> n*n!*hypergeom([2, 1-n], [1, 1], -1):
%p seq(simplify(a(n)), n=0..19); # _Peter Luschny_, Apr 01 2015
%t CoefficientList[Series[(x/(1-x)^2+3*x^2/(1-x)^3+x^3/(1-x)^4)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_ after _Miklos Kristof_, Sep 26 2013 *)
%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(x*(1+x-x^2)*exp(x/(1-x))/(1-x)^4))) \\ _Seiichi Manyama_, Feb 06 2021
%Y Cf. A000578.
%K easy,nonn
%O 0,3
%A _Miklos Kristof_, Apr 13 2005
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