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%I #20 Nov 06 2023 04:19:02
%S 1,0,1,4,41,512,8285,166582,4054953,116797432,3912076929,150190759240,
%T 6532014077809,318632936830136,17286883399233149,1035508343364348938,
%U 68053563847088272945,4879593083836366195728,379847137967853770523937,31960371880691511556886988
%N a(n) = number of labeled graphs on n vertices (with no isolated vertices, multi-edges or loops) such that the degree of every vertex is at most 3.
%C P-recursive.
%D Goulden, I. P.; Jackson, D. M. Labelled graphs with small vertex degrees and $P$-recursiveness. SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093) [Gives e.g.f.]
%H Alois P. Heinz, <a href="/A110041/b110041.txt">Table of n, a(n) for n = 0..150</a>
%F Satisfies the linear recurrence: (-150917976*n^2 - 105258076*n^3 - 1925*n^9 - 13339535*n^5 - 45995730*n^4 - 357423*n^7 - 2637558*n^6 - 120543840*n - n^11 - 66*n^10 - 39916800 - 32670*n^8)*a(n) + (22057180*n^4 + 2*n^10 + 69934280*n^3 + 140581872*n^2 + 161254080*n + 4621890*n^5 + 79833600 + 130*n^9 + 3720*n^8 + 61620*n^7 + 653226*n^6)*a(n + 1) +
%F (3*n^10 + 6932835*n^5 + 5580*n^8 + 92430*n^7 + 979839*n^6 + 241881120*n + 33085770*n^4 + 104901420*n^3 + 210872808*n^2 + 119750400 + 195*n^9)*a(n + 2) + (6932520*n^3 + 39916800 + 136080*n^5 + 24168936*n^2 + 9324*n^6 + 47363040*n + 1223334*n^4 + 6*n^8 + 360*n^7)*a(n + 3) + (6*n^8 + 1431654*n^4 + 372*n^7 + 9996*n^6 + 152040*n^5 + 59875200 + 8545908*n^3 + 31580424*n^2 + 66054960*n)*a(n + 4) + (9100956*n + 6*n^7 + 9646560 + 3631220*n^2 + 335*n^6 + 7929*n^5 + 103085*n^4 + 794709*n^3)*a(n + 5) +
%F (492*n^6 + 9*n^7 + 11032560 + 11359*n^5 + 143385*n^4 + 1067026*n^3 + 4671483*n^2 + 11110486*n)*a(n + 6) + (1021680 + 1041*n^4 + 17838*n^3 + 150699*n^2 + 626358*n + 24*n^5)*a(n + 7) + (461340 + 7027*n^3 + 9*n^5 + 61461*n^2 + 267044*n + 399*n^4)*a(n + 8) + (100980 + 5751*n^2 + 9*n^4 + 39408*n + 372*n^3)*a(n + 9) + (-6414*n - 588*n^2 - 18*n^3 - 23364)*a(n + 10) + (-48*n - 528)*a(n + 11) + 24*a(n + 12) = 0.
%F Differential equation satisfied by the exponential generating function: {F(0) = 1, 9*t^4*(t^4 + t + t^2 - 2)^2*(d^2/dt^2)F(t) + 3*t*(-4*t^6 + 8*t^5 - 16*t + t^10 - 16*t^2 + 2*t^7 + 8 - 2*t^4 + 2*t^8 + 10*t^3)*(t^4 + t + t^2 - 2)*(d/dt)F(t) - t^2*(t^4 + t + t^2 - 2)*(t^10 - 2*t^9 - 6*t^7 - 12*t^6 + t^5 - t^4 + 39*t^3 - 10*t^2 + 24)*F(t)}.
%F Satisfies the recurrence (of order 8): 12*(81*n^4 - 837*n^3 + 2997*n^2 - 4326*n + 1987)*a(n) = 18*(n-1)*(81*n^4 - 810*n^3 + 2709*n^2 - 3435*n + 1036)*a(n-1) + 3*(n-1)*(243*n^6 - 2997*n^5 + 14499*n^4 - 35118*n^3 + 44823*n^2 - 26766*n + 3244)*a(n-2) + 3*(n-2)*(n-1)*(81*n^5 - 1080*n^4 + 4968*n^3 - 9825*n^2 + 7666*n - 178)*a(n-3) + (n-3)*(n-2)*(n-1)*(243*n^5 - 2430*n^4 + 8721*n^3 - 13896*n^2 + 8637*n - 2468)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*(405*n^4 - 3537*n^3 + 11934*n^2 - 15915*n + 6008)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^5 - 2916*n^4 + 11799*n^3 - 19593*n^2 + 11382*n + 502)*a(n-6) + (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(162*n^4 - 1026*n^3 + 2241*n^2 - 1884*n + 182)*a(n-7) - (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 513*n^3 + 972*n^2 - 519*n - 98)*a(n-8). - _Vaclav Kotesovec_, Sep 10 2014
%F a(n) ~ 3^(n/2) * exp(sqrt(3*n) - 3*n/2 - 5/4) * n^(3*n/2) / 2^(n + 1/2) * (1 + 23/(24*sqrt(3*n))). - _Vaclav Kotesovec_, Nov 04 2023, extended Nov 06 2023
%F Limit_{n->oo} A110041(n)/A110040(n) = exp(2). - _Vaclav Kotesovec_, Nov 05 2023
%e Graphs listed by edgeset: a(3) = 4: {(1,2), (2,3)}, {(1,3), (2,3)}, {(1,3), (1,2)}, {(2,3), (1,2), (1,3)}.
%Y Cf. A110040, A110039, A002829, A001205, A001147.
%K easy,nonn
%O 0,4
%A _Marni Mishna_, Jul 08 2005
%E Edited and extended by _Max Alekseyev_, Apr 28 2010
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