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%I #22 Apr 19 2019 01:26:29
%S 0,1,10,54,224,800,2592,7840,22528,62208,166400,433664,1105920,
%T 2768896,6823936,16588800,39845888,94699520,222953472,520486912,
%U 1205862400,2774532096,6343884800,14422114304,32614907904,73400320000
%N a(n) = 2^(n-3)*n^2*(n+3).
%C a(n) is the number of ways to select a subset of {1,2,...n} and then use the subset as an alphabet to form ordered triples.
%D A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F a(n) = Sum_{k=0..n} k^3 * binomial(n, k): binomial transform of A000578.
%F G.f.: x*(1+2*x-2*x^2)/(1-2*x)^4. E.g.f.: x*(1+3*x+x^2)*e^(2*x).
%F A001793(n)*(n+3) = -a(-3-n)*2^(2*n+3) for all n in Z. - _Michael Somos_, Apr 19 2019
%t CoefficientList[Series[(x+3x^2+x^3) Exp[x]^2,{x,0,20}],x] * Table[n!,{n,0,20}]
%o (PARI) a(n)=2^(n-3)*n^2*(n+3)
%Y First differences are in A084903.
%Y Cf. A000578, A001793.
%K nonn,easy
%O 0,3
%A Yong Kong (ykong(AT)curagen.com), Dec 26 2000
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