A002018 From a distribution problem.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Crossrefs

Cf. A000681.

Programs

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 *)

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

Extensions

More terms from David W. Wilson