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%I #13 Jun 21 2022 05:07:41
%S 6,90,1200,15750,211680,2963520,43545600,673596000,10977120000,
%T 188367379200,3399953356800,64457449056000,1281520880640000,
%U 26676557107200000,580481882652672000,13183287756807168000
%N a(n) = n^2*(n+1)*(n+2)!/48.
%D Identity (1.19)/(n+3) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 3.
%H G. C. Greubel, <a href="/A037959/b037959.txt">Table of n, a(n) for n = 2..350</a>
%F (n-1)^2*a(n) = n*(n+2)*(n+1)*a(n-1). - _R. J. Mathar_, Jul 26 2015
%F From _G. C. Greubel_, Jun 20 2022: (Start)
%F a(n) = (1/(n+3))*Sum_{j=0..n} (-1)^(n+j)*binomial(n,j)*j^(n+3).
%F a(n) = n!*StirlingS2(n+3, n)/(n+3).
%F a(n) = A037961(n)/(n+3).
%F a(n) = A131689(n+3, n).
%F a(n) = A019538(n+3, n).
%F E.g.f.: x*(1 + 6*x + 3*x^2)/(4*(1-x)^6). (End)
%t Table[(n+2)!n^2(n+1)/48,{n,2,20}] (* _Harvey P. Dale_, Jul 29 2021 *)
%o (Magma) [Factorial(n)*StirlingSecond(n+3,n)/(n+3): n in [2..30]]; // _G. C. Greubel_, Jun 20 2022
%o (SageMath) [factorial(n)*stirling_number2(n+3, n)/(n+3) for n in (2..30)] # _G. C. Greubel_, Jun 20 2022
%Y Cf. A000142, A001297, A019538, A131689.
%Y Cf. A037960, A037961, A037962, A037963.
%K nonn
%O 2,1
%A _N. J. A. Sloane_
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