A324311 Expansion of the unique formal power series R(t) with constant term 0 satisfying t = Sum_{n>=0} (1/(n+1))*binomial(2n,n)^2/*R(t)^(n+1).

1, -2, -4, -20, -132, -1008, -8432, -75096, -700180, -6761040, -67116048, -681341440, -7045987312, -74007446400, -787712891328, -8480626018544, -92218188224340, -1011605255827920, -11183503253443920, -124495464358157760, -1394538057040652656, -15708893392461609600

Offset

1,2

Links

Table of n, a(n) for n=1..22.

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 1.

Mathematica

m = 22;
aa = Array[a, m]; R[t_] = aa.t^Range[m]; eq = Thread[CoefficientList[t-Sum[ 1/(n+1) Binomial[2n, n]^2 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)^2/(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. A324312, A324313, A324314.

Sequence in context: A325617 A013329 A364393 * A303671 A330663 A370766

Adjacent sequences: A324308 A324309 A324310 * A324312 A324313 A324314

Keyword

sign

Author

Michel Marcus, Feb 21 2019

Status

approved