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%I M4985 #62 Nov 23 2022 08:57:37
%S 1,15,180,2100,25200,317520,4233600,59875200,898128000,14270256000,
%T 239740300800,4249941696000,79332244992000,1556132497920000,
%U 32011868528640000,689322235650048000,15509750302126080000,364022962973429760000,8898339094906060800000
%N Number of simplices in barycentric subdivision of n-simplex.
%D R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.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 G. C. Greubel, <a href="/A005461/b005461.txt">Table of n, a(n) for n = 1..440</a>
%H R. Austin, R. K. Guy, and R. Nowakowski, <a href="/A000629/a000629.pdf">Unpublished notes, 1987</a>.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">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, Vol. 10, No. 7 (2022), 1161.
%F a(n) = n*(n + 1)*(n + 3)!/48.
%F Essentially Stirling numbers of second kind - see A028246.
%F If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-3) = (-1)^n*f(n,4,-3), (n>=4). - _Milan Janjic_, Mar 01 2009
%F E.g.f.: t*(3*t + 2)/(2*(t - 1)^6). - _Ran Pan_, Jul 10 2016
%F a(n) ~ sqrt(Pi/2)*exp(-n)*n^(n+1/2)*(n^5/24 + 85*n^4/288 + 5065*n^3/6912 + 955841*n^2/1244160 + 3710929*n/11943936). - _Ilya Gutkovskiy_, Jul 10 2016
%F From _Amiram Eldar_, May 06 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 16*(e + gamma - Ei(1)) - 64/3, where e = A001113, gamma = A001620, and Ei(1) = A091725.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 32*(gamma - Ei(-1)) - 16/e - 56/3, where Ei(-1) = -A099285. (End)
%F a(n) = (n-1)! * Stirling2(n+3, n). - _G. C. Greubel_, Nov 23 2022
%e G.f. = x + 15*x^2 + 180*x^3 + 2100*x^4 + 25200*x^5 + 317520*x^6 + ...
%p a:=n->sum((n-j)*n!/4!, j=3..n): seq(a(n), n=4..17); # _Zerinvary Lajos_, Apr 29 2007
%t Table[(n(n+1)(n+3)!)/48,{n,20}] (* _Harvey P. Dale_, Mar 14 2012 *)
%t a[ n_] := If[ n < 0, 0, n (n + 1) (n + 3)! / 48]; (* _Michael Somos_, May 27 2014 *)
%o (Sage) [factorial(m+1)*binomial(m-1,2)/24 for m in range(3, 19)] # _Zerinvary Lajos_, Jul 05 2008
%o (Sage) [binomial(n,4)*factorial (n-2)/2 for n in range(4, 18)] # _Zerinvary Lajos_, Jul 07 2009
%o (Magma) [Factorial(n-1)*StirlingSecond(n+3,n): n in [1..35]]; // _G. C. Greubel_, Nov 23 2022
%Y Cf. A001113, A001620, A028246, A091725, A099285.
%Y Cf. A005460, A005462, A005463, A005464, A005465.
%Y Cf. A001297.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Harvey P. Dale_, Mar 14 2012
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