OFFSET
1,4
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..303
Zhi-Cheng Gao, The number of rooted 2-connected triangular maps on the projective plane, Journal of Combinatorial Theory, Series B, Volume 53, Issue 1, September 1991, Pages 130-142.
FORMULA
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)
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.
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
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
PROG
(PARI)
seq(N) = {
my(x = 'x + O('x^(N+1)), r=serreverse(x*(1-2*x)^2),
v = Vec(subst((1-sqrt((1-6*x)/(1-2*x)))/(2*x)-1/(1-3*x), 'x, r)));
concat([0, 0], v);
};
seq(21)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Jul 10 2018
STATUS
approved