%I #46 May 26 2023 01:03:05
%S 1,1,35,5775,2627625,2546168625,4509264634875,13189599057009375,
%T 59287247761257140625,388035036597427985390625,
%U 3546252199463894358484921875,43764298393583920278062420859375,709638098451963267308782154234765625,14778213400262135041705388361938994140625
%N Number of partitions of { 1, 2, ..., 4n } into sets of size 4.
%C P-recursive. - _Marni Mishna_, Jul 11 2005
%H Andrew Howroyd, <a href="/A025036/b025036.txt">Table of n, a(n) for n = 0..50</a>
%H Cyril Banderier, Philippe Marchal, and Michael Wallner, <a href="https://arxiv.org/abs/1805.09017">Rectangular Young tableaux with local decreases and the density method for uniform random generation</a> (short version), arXiv:1805.09017 [cs.DM], 2018.
%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 17.
%F a(n) = (4n)!/(n!(4!)^n). - _Christian G. Bower_, Sep 15 1998
%F 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
%F 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
%F E.g.f.: exp(x^4/4!) (with interpolated zeros). - _Paul Barry_, May 26 2003
%F a(n) = Pochhammer(n+1, 3*n)/24^n. - _Peter Luschny_, Nov 18 2019
%e a(1)=1: {1,2,3,4}.
%e One of the a(2)=35 partitions for n = 8: {1,2,3,4}{5,6,7,8}.
%p a := pochhammer(n + 1, 3*n) / 24^n:
%p seq(a(n), n=0..13); # _Peter Luschny_, Nov 18 2019
%t 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_ *)
%Y Column k=4 of A060540.
%Y Cf. A025035, A110103, A002829.
%K nonn,easy
%O 0,3
%A _David W. Wilson_
%E Edited by _N. J. A. Sloane_, Aug 23 2008 at the suggestion of _R. J. Mathar_