A316698 - OEIS

%I #16 Jul 11 2018 09:44:02

%S 0,0,1,18,261,3539,46695,608526,7884661,101905839,1316047599,

%T 16998339587,219699143367,2842235616645,36809980380883,

%U 477280717428102,6195737611180053,80522713890559319,1047702563499718623,13646946767000964471,177947654115176898479

%N a(n) is the number of rooted 2-connected triangular maps on the projective plane with n vertices.

%H Gheorghe Coserea, <a href="/A316698/b316698.txt">Table of n, a(n) for n = 1..303</a>

%H Zhi-Cheng Gao, <a href="https://doi.org/10.1016/0095-8956(91)90058-R">The number of rooted 2-connected triangular maps on the projective plane</a>, Journal of Combinatorial Theory, Series B, Volume 53, Issue 1, September 1991, Pages 130-142.

%F G.f. A(x) = (1 - sqrt((1-6*r)/(1-2*r)))/(2*r) - 1/(1-3*r), where r(x) satisfies x = r*(1-2*r)^2, with r(0)=0. (see (1.1) in Gao link)

%F G.f. y=A(x) satisfies: 0 = (729*x^2 - 54*x + 1)*y^6 + (-567*x^2 + 48*x - 1)*y^5 + (4617*x^3 - 486*x^2 + 12*x)*y^4 + (-14310*x^4 + 1772*x^3 - 54*x^2)*y^3 + (-672*x^4 + 50*x^3)*y^2 + (126*x^5 - 36*x^4 + 2*x^3)*y - 2*x^6.

%F Recurrence: (n-1)*n*(2*n - 3)*(4*n - 9)*(4*n - 3)*(972*n^7 - 1944*n^6 - 169443*n^5 + 1865607*n^4 - 8817457*n^3 + 21764795*n^2 - 27508222*n + 14065464)*a(n) = 3*(n-1)*(699840*n^11 - 5598720*n^10 - 107581284*n^9 + 2120974416*n^8 - 16716827583*n^7 + 77044659801*n^6 - 229110154570*n^5 + 453176543549*n^4 - 592757452327*n^3 + 491840891214*n^2 - 233773288056*n + 48250762560)*a(n-1) - 9*(5668704*n^12 - 61410960*n^11 - 770480100*n^10 + 20379495348*n^9 - 192680893665*n^8 + 1066797111051*n^7 - 3886131103119*n^6 + 9712411159089*n^5 - 16796662782944*n^4 + 19765806847064*n^3 - 15086450010036*n^2 + 6716653116768*n - 1318624045200)*a(n-2) + 486*(3*n - 11)*(3*n - 10)*(122472*n^10 - 717336*n^9 - 21548106*n^8 + 353617272*n^7 - 2470176720*n^6 + 10020300957*n^5 - 25599297354*n^4 + 41773597853*n^3 - 42167708852*n^2 + 23887121874*n - 5766718860)*a(n-3) - 26244*(n-4)*(3*n - 14)*(3*n - 13)*(3*n - 11)*(3*n - 10)*(972*n^7 + 4860*n^6 - 160695*n^5 + 1023252*n^4 - 3054319*n^3 + 4802888*n^2 - 3820650*n + 1199772)*a(n-4). - _Vaclav Kotesovec_, Jul 11 2018

%F a(n) ~ (27/2)^n * (1/(2*3^(7/4)*Gamma(3/4)) - 10/(27*sqrt(3*Pi)*n^(1/4)) + sqrt(2)*Gamma(3/4) / (3^(9/4)*Pi*sqrt(n))) / n^(5/4) [main asymptotic term by Gao, 1991]. - _Vaclav Kotesovec_, Jul 11 2018

%o (PARI)

%o seq(N) = {

%o   my(x = 'x + O('x^(N+1)), r=serreverse(x*(1-2*x)^2),

%o      v = Vec(subst((1-sqrt((1-6*x)/(1-2*x)))/(2*x)-1/(1-3*x), 'x, r)));

%o   concat([0,0], v);

%o };

%o seq(21)

%Y Cf. A007137, A316598.

%K nonn

%O 1,4

%A _Gheorghe Coserea_, Jul 10 2018