A025036 Number of partitions of { 1, 2, ..., 4n } into sets of size 4.

1, 1, 35, 5775, 2627625, 2546168625, 4509264634875, 13189599057009375, 59287247761257140625, 388035036597427985390625, 3546252199463894358484921875, 43764298393583920278062420859375, 709638098451963267308782154234765625, 14778213400262135041705388361938994140625

Offset

0,3

Comments

P-recursive. - Marni Mishna, Jul 11 2005

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 17.

Formula

a(n) = (4n)!/(n!(4!)^n). - Christian G. Bower, Sep 15 1998

E.g.f.: A(t) = Sum a(n)*t^(4n)/(4n!) = exp(t^4/4!); recurrence: 3*a(n) - (4*n-3)*(2*n-1)*(4*n-1)*a(n-1) = 0. - Marni Mishna, Jul 11 2005

Integral representation as n-th moment of a positive function on the positive axis in Maple notation: a(n)=int(x^n*(1/4*(2^(3/4)*hypergeom([], [5/4, 3/2], -3/32*x)*3^(3/4)*GAMMA(3/4)^2*x*Pi^(1/2)-2*hypergeom([], [3/4, 5/4], -3/32*x)*3^(1/2)*2^(1/2)*Pi*x^(3/4)*GAMMA(3/4)+hypergeom([], [1/2, 3/4], -3/32*x)*3^(1/4)*2^(3/4)*Pi^(3/2)*x^(1/2))/Pi^(3/2)/x^(5/4)/GAMMA(3/4)), x=0..infinity), n=0, 1..., with offset 1. -Karol A. Penson, Oct 06 2005

E.g.f.: exp(x^4/4!) (with interpolated zeros). - Paul Barry, May 26 2003

a(n) = Pochhammer(n+1, 3*n)/24^n. - Peter Luschny, Nov 18 2019

Example

a(1)=1: {1,2,3,4}.
One of the a(2)=35 partitions for n = 8: {1,2,3,4}{5,6,7,8}.

Maple

a := pochhammer(n + 1, 3*n) / 24^n:
seq(a(n), n=0..13); # Peter Luschny, Nov 18 2019

Mathematica

terms = 12; max = 4*(terms-1); DeleteCases[CoefficientList[Exp[x^4/4!] + O[x]^(max+1), x]*Range[0, max]!, 0] (* Jean-François Alcover, Jun 29 2018, after Paul Barry *)

Crossrefs

Column k=4 of A060540.

Cf. A025035, A110103, A002829.

Sequence in context: A336306 A202881 A210268 * A224126 A249888 A212025

Adjacent sequences: A025033 A025034 A025035 * A025037 A025038 A025039

Keyword

nonn,easy

Author

David W. Wilson

Extensions

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

Status

approved