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%I #13 Jan 31 2019 19:34:55
%S 1,0,75,122220,757275750,12713292692100,474415445827323000,
%T 34461884930947363890000,4431555785100983345799993000,
%U 939388724430508823324694340500000
%N 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.)
%C P-recursive
%H Denis S. Krotov, Konstantin V. Vorob'ev, <a href="https://arxiv.org/abs/1812.02166">On unbalanced Boolean functions attaining the bound 2n/3-1 on the correlation immunity</a>, arXiv:1812.02166 [math.CO], 2018.
%H Marni Mishna, <a href="https://web.archive.org/web/20070623105521/http://algo.inria.fr/mishna/Reg-Asympt/reg-asympt.html">Maple worksheet</a>
%F 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}.
%F 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)}.
%F a(n) ~ 3^(4*n+1/2) * n^(4*n) / (2^n * exp(4*n+1)). - _Vaclav Kotesovec_, Mar 11 2014
%e One of the 75 2-regular 3-hypergraphs on 6 vertices: {1,2,3} {4,5,6} {1,2,4} {3,5,6}.
%Y Cf. A025035, A110101, A110103, A001205.
%K easy,nonn
%O 0,3
%A _Marni Mishna_, Jul 11 2005
%E Replaced broken link, _Vaclav Kotesovec_, Mar 11 2014
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