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A005461 Number of simplices in barycentric subdivision of n-simplex.
(Formerly M4985)
12
1, 15, 180, 2100, 25200, 317520, 4233600, 59875200, 898128000, 14270256000, 239740300800, 4249941696000, 79332244992000, 1556132497920000, 32011868528640000, 689322235650048000, 15509750302126080000, 364022962973429760000, 8898339094906060800000 (list; graph; refs; listen; history; text; internal format)
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R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.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
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987.
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. A001297.
Sequence in context: A293476 A004992 A055084 * A373759 A138443 A235455
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Mar 14 2012
STATUS
approved

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Last modified July 25 08:50 EDT 2024. Contains 374587 sequences. (Running on oeis4.)