A005461 Number of simplices in barycentric subdivision of n-simplex.

1, 15, 180, 2100, 25200, 317520, 4233600, 59875200, 898128000, 14270256000, 239740300800, 4249941696000, 79332244992000, 1556132497920000, 32011868528640000, 689322235650048000, 15509750302126080000, 364022962973429760000, 8898339094906060800000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

			G.f. = x + 15*x^2 + 180*x^3 + 2100*x^4 + 25200*x^5 + 317520*x^6 + ...
		

References

• R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
• R. K. Guy, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..440
• R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics, Vol. 10, No. 7 (2022), 1161.

Crossrefs

Cf. A001113, A001620, A028246, A091725, A099285.

Cf. A005460, A005462, A005463, A005464, A005465.

Cf. A001297.

Programs

Magma

[Factorial(n-1)*StirlingSecond(n+3,n): n in [1..35]]; // G. C. Greubel, Nov 23 2022

Maple

a:=n->sum((n-j)*n!/4!, j=3..n): seq(a(n), n=4..17); # Zerinvary Lajos, Apr 29 2007

Mathematica

Table[(n(n+1)(n+3)!)/48,{n,20}] (* Harvey P. Dale, Mar 14 2012 *)
a[ n_] := If[ n < 0, 0, n (n + 1) (n + 3)! / 48]; (* Michael Somos, May 27 2014 *)

Sage

[factorial(m+1)*binomial(m-1,2)/24 for m in range(3, 19)] # Zerinvary Lajos, Jul 05 2008

Sage

[binomial(n,4)*factorial (n-2)/2 for n in range(4, 18)] #  Zerinvary Lajos, Jul 07 2009

Formula

a(n) = n*(n + 1)*(n + 3)!/48.

Essentially Stirling numbers of second kind - see A028246.

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

E.g.f.: t*(3*t + 2)/(2*(t - 1)^6). - Ran Pan, Jul 10 2016

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

From Amiram Eldar, May 06 2022: (Start)

Sum_{n>=1} 1/a(n) = 16*(e + gamma - Ei(1)) - 64/3, where e = A001113, gamma = A001620, and Ei(1) = A091725.

Sum_{n>=1} (-1)^(n+1)/a(n) = 32*(gamma - Ei(-1)) - 16/e - 56/3, where Ei(-1) = -A099285. (End)

a(n) = (n-1)! * Stirling2(n+3, n). - G. C. Greubel, Nov 23 2022

Extensions

More terms from Harvey P. Dale, Mar 14 2012