|
%I #11 Dec 25 2017 03:49:03
%S 1,0,0,0,1,12,330,11205,505505,28787052,2024844444,172592502570,
%T 17545270969545,2098273032696720,291739927315433454,
%U 46676360010342811203
%N a(n) is the number of 3-regular 3-hypergraphs on n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 3-regular implies that each vertex is in exactly 3 hyperedges.)
%C P-recursive
%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 Differential equation satisfied by exponential generating function: {F(0) = 1, 36*t^2*(t + 1)*(t^2 - 2)^2*(3*t^2 + 2*t - 2)^2*(d^2/dt^2)F(t) - 12*(t + 1)*(3*t^2 + 2*t - 2)*(3*t^10 + 2*t^9 - 8*t^8 - 40*t^7 - 56*t^6 + 4*t^5 - 48*t^4 - 96*t^3 + 80*t^2 + 80*t - 32)*(d/dt)F(t) + t^3*(t + 1)*(3*t^2 + 2*t - 2)*(3*t^9 + 2*t^8 - 2*t^7 - 108*t^6 - 144*t^5 + 32*t^4 - 24*t^3 + 16*t^2 + 112*t - 64)*F(t)}.
%F Linear recurrence for a(n): initial values: a(2) = 0, a(3) = 0, a(0) = 1, a(1) = 0, a(4) = 1, a(5) = 12, a(6) = 330, a(7) = 11205, a(8) = 505505, a(9) = 28787052, a(10) = 2024844444, a(11) = 172592502570, a(12) = 17545270969545;
%F then (1971620508*n^4 + 4242044664*n^3 + 3*n^12 + 4459328640*n + 1437004800 +
%F 167310*n^9 + 5794678656*n^2 + 20779902*n^7 + 234*n^11 + 8151*n^10 + 2248389*n^8
%F + 618210450*n^5 + 134970693*n^6)*a(n) + (154*n^10 + 77519860*n^5 + 334620440*n^4
%F + 958003200 + 5280*n^9 + 106260*n^8 + 1392666*n^7 + 12460602*n^6 + 979793232*n^3
%F + 1848236544*n^2 + 2014882560*n + 2*n^11)*a(n + 1) + ( - 96300*n^7 - 1200066*n^6
%F - 540148032*n^2 - 767940480*n - 4980*n^8 - 57398920*n^4 - 219822600*n^3
%F - 479001600 - 10060470*n^5 - 2*n^10 - 150*n^9)*a(n + 2) + ( - 97416*n^8
%F - 17244057600 - 24771847680*n - 2808*n^9 - 36*n^10 - 1978992*n^7 - 26064612*n^6
%F - 232501752*n^5 - 1422206064*n^4 - 5889271968*n^3 - 15795689472*n^2)*a(n
%F + 3) + ( - 5364230400*n - 4790016000 - 24*n^9 - 1872*n^8 - 64368*n^7 - 1280160*n^6
%F - 16223256*n^5 - 135808848*n^4 - 750702432*n^3 - 2641118400*n^2)*a(n + 4)
%F + (3252704*n^5 + 2043740160 + 194208*n^6 + 2058817536*n + 33702144*n^4 +
%F 221164160*n^3 + 897495552*n^2 + 6560*n^7 + 96*n^8)*a(n + 5) + (246432*n^6
%F + 48931572*n^4 + 4055546880 + 1512709248*n^2 + 4406952*n^5 + 7824*n^7 +
%F 345350856*n^3 + 108*n^8 + 3758813568*n)*a(n + 6) + (528439296*n + 2696360*n^4
%F + 27036368*n^3 + 161115712*n^2 + 159784*n^5 + 5208*n^6 + 72*n^7 + 735989760)*a(n
%F + 7) + ( - 59595808*n^2 - 8517816*n^3 - 338532480 - 504*n^6 - 680168*n^4
%F - 220837728*n - 28776*n^5)*a(n + 8) + ( - 262432*n^3 - 288*n^5 - 11355392*n
%F - 13824*n^4 - 20613120 - 2459328*n^2)*a(n + 9) + (31392*n^3 + 3713184*n
%F + 720*n^4 + 512496*n^2 + 10074240)*a(n + 10) + (253440 + 288*n^3 + 8544*n^2
%F + 82176*n)*a(n + 11) + ( - 7584*n - 49536 - 288*n^2)*a(n + 12) + 384*a(n + 13).
%F a(n) ~ n^(2*n) * 3^(n+1/2) / (exp(2*n+2) * 4^n). - _Vaclav Kotesovec_, Mar 11 2014
%F Recurrence (of order 11): 192*(243*n^2 - 285*n - 290)*a(n) = 144*(n-1)*(243*n^3 - 285*n^2 + 34*n + 796)*a(n-1) + 48*(n-2)*(n-1)*(1701*n^2 - 24*n + 1027)*a(n-2) - 48*(n-3)*(n-2)*(n-1)*(729*n^3 - 2556*n^2 - 2601*n + 2558)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*(3645*n^3 - 7110*n^2 - 35091*n + 30676)*a(n-4) + 12*(n-4)*(n-3)*(n-2)*(n-1)*(729*n^4 - 4500*n^3 + 1623*n^2 + 12924*n - 11872)*a(n-5) + 8*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(729*n^3 - 4338*n^2 - 1728*n + 3269)*a(n-6) - 8*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(486*n^2 + 2145*n - 3485)*a(n-7) - 12*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^3 - 1014*n^2 - 1304*n + 1619)*a(n-8) + 24*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(18*n - 13)*a(n-9) - 6*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(27*n - 71)*a(n-10) + (n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^2 + 201*n - 332)*a(n-11). - _Vaclav Kotesovec_, Mar 11 2014
%e The 3-regular 3-hypergraphs on 4 vertices: {1,2,3}, {2,3,4},{3,4,1},{4,1,2}.
%Y Cf. A025035, A110100, A002829.
%K easy,nonn
%O 0,6
%A _Marni Mishna_, Jul 11 2005
%E Replaced broken link, _Vaclav Kotesovec_, Mar 11 2014
| 