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%I M4433 #44 Nov 22 2022 22:18:26
%S 1,7,50,390,3360,31920,332640,3780000,46569600,618710400,8821612800,
%T 134399865600,2179457280000,37486665216000,681734237184000,
%U 13071512982528000,263564384219136000,5575400435404800000,123469776914964480000,2856835183101419520000
%N a(n) = (3*n+4)*(n+3)!/24.
%C Essentially Stirling numbers of second kind - third external diagonal of Worpitzky triangle A028246.
%D R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
%D R. K. Guy, personal communication.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005460/b005460.txt">Table of n, a(n) for n = 0..300</a>
%H R. Austin, R. K. Guy, and R. Nowakowski, <a href="/A000629/a000629.pdf">Unpublished notes, 1987</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.
%H John K. Sikora, <a href="https://arxiv.org/abs/1806.00887">On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles</a>, arXiv:1806.00887 [math.NT], 2018.
%F E.g.f.: (1+2*x)/(1-x)^5.
%F a(n) = S2(n+3, n+1)*n! = n!*A001296(n+1). - _Olivier Gérard_, Sep 13 2016
%t Table[StirlingS2[n+3, n+1]*n!, {n,0,20}]
%o (Magma) [(3*n+4)*Factorial(n+3)/24: n in [0..20]]; // _Vincenzo Librandi_, Oct 08 2011
%o (PARI) a(n)=(3*n+4)*(n+3)!/24 \\ _Charles R Greathouse IV_, Jun 30 2017
%o (SageMath) [factorial(n)*stirling_number2(n+3,n+1) for n in range(21)] # _G. C. Greubel_, Nov 22 2022
%Y Cf. A028246.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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