A002018 From a distribution problem. (Formerly M3652 N1485)

1, 1, 4, 33, 480, 11010, 367560, 16854390, 1016930880, 78124095000, 7446314383200, 862332613342200, 119261328828364800, 19415283189746043600, 3675162134109650184000, 800409618620667941886000, 198730589981586780813696000, 55800304882692417053710704000

Offset

0,3

References

H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..253

Formula

a(n) = n*(2*n-1)*b(n-1) - n*(n-1)^2*b(n-2), b(n) = A000681(n). [corrected by Seiichi Manyama, Apr 22 2025]

From Seiichi Manyama, Apr 22 2025: (Start)

a(n) = (n-1)! * n! * Sum_{k=0..n-1} (-1)^k * (1/2)^(n-k-1) * binomial(-3/2,k)/(n-k-1)! for n > 0.

a(n) = (n-1)! * n! * [x^(n-1)] 1/(1-x)^(3/2) * exp(x/2) for n > 0.

a(n) = n * ( n*a(n-1) - (n-1)*(n-2)/2 * a(n-2) ) for n > 1. (End)

a(n) ~ 4 * sqrt(Pi) * n^(2*n + 1/2) / exp(2*n - 1/2). - Vaclav Kotesovec, Apr 24 2025

Mathematica

b[n_] := Sum[(2i)!*n!^2/(2^i*i!^2*(n-i)!), {i, 0, n}]/2^n; a[n_] := n*(2n-1)*b[n-1] - n*(n-1)^2*b[n-2]; a[0]=1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 08 2012, after formula *)

Crossrefs

Cf. A000681.

Sequence in context: A331690 A002190 A101981 * A368837 A219504 A258180

Adjacent sequences: A002015 A002016 A002017 * A002019 A002020 A002021

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Status

approved