A324313 Expansion of the unique formal power series R(t) with constant term 0 satisfying t = Sum_{n>=0} (1/(n+1))*binomial(2n,n)*binomial(3n,n)/*R(t)^(n+1).

1, -3, -12, -105, -1206, -16002, -232416, -3592377, -58113000, -973372686, -16757906256, -295003183410, -5289613240464, -96324638802300, -1777361586269760, -33170533400455353, -625219596117283590, -11887661990007138006, -227778723332612670600, -4394621278195058676150

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Mireille Bousquet-Mélou, Andrew Elvey Price, Andrew Price, The generating function of planar Eulerian orientations, arXiv:1803.08265 [math.CO], 2018.

Mireille Bousquet-Mélou, Andrew Elvey Price, Paul Zinn-Justin, Eulerian orientations and the six-vertex model on planar map, arXiv:1902.07369 [math.CO], 2019. See Theorem 2.

Mathematica

m = 20;
aa = Array[a, m];
R[t_] = aa.t^Range[m];
eq = Thread[CoefficientList[t-Sum[1/(n+1) Binomial[2n, n] Binomial[3n, n]* R[t]^(n+1), {n, 0, m}] + O[t]^(m+1), t] == 0];
aa /. Solve[eq, aa][[1]] (* Jean-François Alcover, Feb 25 2019 *)

Prog

(PARI) lista(nn) = {my(v = vector(nn), R, P, c, r, s); kill(y); for (n=1, nn, v[n] = y; R = sum(k=1, n, v[k]*t^k); P = sum(k=0, n, binomial(2*k, k)*binomial(3*k, k)/(k+1)*R^(k+1)); c = polcoef(P, n, t); r = polcoef(c, 0, y); s = polcoef(c, 1, y); if (n==1, v[n] = (1-r)/s, v[n] = -r/s); ); R = sum(k=1, #v, v[k]*t^k); vector(nn, k, polcoef(R, k, t)); }

Crossrefs

Cf. A324311, A324312, A324314.

Sequence in context: A350953 A366668 A141535 * A111485 A293126 A080447

Adjacent sequences: A324310 A324311 A324312 * A324314 A324315 A324316

Keyword

sign

Author

Michel Marcus, Feb 21 2019

Status

approved