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%I #17 Sep 03 2019 20:21:05
%S 0,2,42,892,18888,399280,8431776,177936064,3753206400,79139040000,
%T 1668268861952,35160393493504,740921108899840,15611120289755136,
%U 328889518650990592,6928313584957702144,145939409585973133312,3073901537848967495680,64741608434203590524928
%N Number of essentially 3-connected rooted toroidal maps with n vertices.
%H Michael De Vlieger, <a href="/A308526/b308526.txt">Table of n, a(n) for n = 0..500</a>
%H Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, <a href="https://arxiv.org/abs/1907.04016">A bijection for essentially 3-connected toroidal maps</a>, arXiv:1907.04016 [math.CO], 2019.
%F G.f.: (1+A)*(A^2+3*A+4)*A/((3*A^2+2*A-2)^2*(A+2)) where A=x*(2+2*A+A^2)^2.
%F From _Vaclav Kotesovec_, Jun 25 2019: (Start)
%F Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(20802264*n^8 - 513044308*n^7 + 5457802931*n^6 - 32703730375*n^5 + 120697828661*n^4 - 280851750277*n^3 + 402186188144*n^2 - 323841737040*n + 112137480000)*a(n) = 6*(96792934392*n^11 - 2657166947316*n^10 + 32322659739783*n^9 - 229681592172541*n^8 + 1057798736706708*n^7 - 3309904792738002*n^6 + 7166955104700747*n^5 - 10716261762345309*n^4 + 10816142222455650*n^3 - 6994792735444832*n^2 + 2594496776694720*n - 413761340160000)*a(n-1) - 32*(136213224672*n^11 - 3864805132664*n^10 + 48853431813424*n^9 - 362854015235883*n^8 + 1757540351761182*n^7 - 5820283983972594*n^6 + 13419220917200106*n^5 - 21479458450012897*n^4 + 23298284090559356*n^3 - 16214747993479962*n^2 + 6458737193497260*n - 1099216619550000)*a(n-2) - 768*(29622423936*n^11 - 845570009984*n^10 + 10735773789272*n^9 - 79940670306164*n^8 + 387373872945691*n^7 - 1280558339496068*n^6 + 2940763323423808*n^5 - 4679130395980206*n^4 + 5037190265229413*n^3 - 3476169558457578*n^2 + 1372907413337880*n - 231844115160000)*a(n-3) - 24576*(2*n - 7)*(582463392*n^10 - 14885297224*n^9 + 166341178864*n^8 - 1068833075597*n^7 + 4366030094616*n^6 - 11823901892456*n^5 + 21447449277486*n^4 - 25646549248003*n^3 + 19256170722842*n^2 - 8132937809520*n + 1445811660000)*a(n-4) - 131072*(n-5)*(2*n - 9)*(2*n - 7)*(20802264*n^8 - 346626196*n^7 + 2448956167*n^6 - 9565916473*n^5 + 22545828451*n^4 - 32733304759*n^3 + 28456182418*n^2 - 13430023272*n + 2589840000)*a(n-5).
%F a(n) ~ (7 + sqrt(7)) * 2^(4*n - 5) * (17 + 7*sqrt(7))^n / 3^(3*n + 1).
%F (End)
%p dev_A := 0; n := 20; dev_A := series(RootOf(A-x*(A^2+2*A+2)^2, A), x = 0, n+1): seq(coeff(series(subs(A = dev_A, (1+A)*(A^2+3*A+4)*A/((3*A^2+2*A-2)^2*(A+2))), x, n+1), x, k), k = 0 .. n);
%t Block[{nn = 19, A, x}, A[_] = 0; Do[A[x_] = x*(2 + 2*A[x] + A[x]^2)^2 + O[x]^nn, nn]; CoefficientList[(1 + A[x])*(A[x]^2 + 3*A[x] + 4)* A[x]/((3*A[x]^2 + 2*A[x] - 2)^2*(A[x] + 2)), x]] (* _Michael De Vlieger_, Sep 03 2019 *)
%Y Cf. A308523, A308524, A289208, A006422.
%K nonn
%O 0,2
%A _Nicolas Bonichon_, Jun 05 2019
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