A005461 Number of simplices in barycentric subdivision of n-simplex. (Formerly M4985)

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

Offset

1,2

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R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.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R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987.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.

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

Example

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

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 *)

Prog

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

Crossrefs

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

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

Cf. A001297.

Sequence in context: A293476 A004992 A055084 * A373759 A138443 A235455

Adjacent sequences: A005458 A005459 A005460 * A005462 A005463 A005464

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Mar 14 2012

Status

approved