A110103 -id:A110103 - cp's OEIS frontend

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

David W. Wilson

P-recursive. - Marni Mishna, Jul 11 2005

			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}.

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.

Column k=4 of A060540.

Cf. A025035, A110103, A002829.

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

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 *)

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
a(n) ~ 2^(5*n+1) * (n/e)^(3*n) / 3^n. - Amiram Eldar, Aug 28 2025

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

A110100 a(n) is the number of 2-regular 3-hypergraphs on 3n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 2-regular implies that each vertex is in exactly 2 hyperedges.)

Original entry on oeis.org

1, 0, 75, 122220, 757275750, 12713292692100, 474415445827323000, 34461884930947363890000, 4431555785100983345799993000, 939388724430508823324694340500000

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive

			One of the 75 2-regular 3-hypergraphs on 6 vertices: {1,2,3} {4,5,6} {1,2,4} {3,5,6}.

Denis S. Krotov, Konstantin V. Vorob'ev, On unbalanced Boolean functions attaining the bound 2n/3-1 on the correlation immunity, arXiv:1812.02166 [math.CO], 2018.
Marni Mishna, Maple worksheet

Cf. A025035, A110101, A110103, A001205.

Recurrence: {a(0) = 1, a(1) = 0, (361631520*n + 1358261784*n^2 + 2841968052*n^3 + 3241507005*n^5 + 3725654130*n^4 + 1922779782*n^6 + 781684101*n^7 + 214347870*n^8 + 37889775*n^9 + 3897234*n^10 + 177147*n^11 + 39916800)*a(n) + (870112800*n + 1655958600*n^2 + 1805971896*n^3 + 561697416*n^5 + 1244162430*n^4 + 166255740*n^6 + 31125384*n^7 + 3346110*n^8 + 157464*n^9 + 199584000)*a(n + 1) + (70976400*n + 86362056*n^2 + 57212568*n^3 + 5161320*n^5 + 22352760*n^4 + 653184*n^6 + 34992*n^7 + 24393600)*a(n + 2) + (-468192*n-411840-198432*n^2-37152*n^3-2592*n^4)*a(n + 3) + 64*a(n + 4), a(2) = 75, a(3) = 122220}.
Differential equation satisfied by generating series A(t)=sum a(n) t^(3n)/(3n)!: {F(0) = 1, 16*t^5*(-2 + t^3)^3*(d^2/dt^2)F(t) + 8*t*(t^9-20*t^3 + 8)*(-2 + t^3)^2*(d/dt)F(t) + t^6*(t^3 + 10)*(t^3-4)*(-2 + t^3)^2*F(t)}.
a(n) ~ 3^(4*n+1/2) * n^(4*n) / (2^n * exp(4*n+1)). - Vaclav Kotesovec, Mar 11 2014

Replaced broken link, Vaclav Kotesovec, Mar 11 2014

cp's OEIS Frontend

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

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