A349310 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^4) / (1 - x).

1, 2, 10, 74, 642, 6082, 60970, 635818, 6826690, 74958914, 837833482, 9500939978, 109037364930, 1264049402754, 14780619799722, 174121322204074, 2064572904600706, 24620095821589378, 295087003429677322, 3552841638851183690, 42950428996378731010

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(4*k,k) / (3*k+1).

a(n) = F([(1+n)/3, (2+n)/3, (3+n)/3, -n], [2/3, 1, 4/3], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021

a(n) ~ sqrt(1 + 3*r) / (2^(13/6) * sqrt(3*Pi) * (1-r)^(1/6) * n^(3/2) * r^(n + 1/3)), where r = 0.0766602099042102089064087954661556186872273232742446843... is the smallest real root of the equation 3^3 * (1-r)^4 = 4^4 * r. - Vaclav Kotesovec, Nov 15 2021

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = (1 + x A[x]^4)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 3 k, 4 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]

Crossrefs

Cf. A002293, A006318, A346626, A346646, A349311, A349312, A349313, A349314.

Sequence in context: A001395 A047853 A151387 * A192259 A245901 A352914

Adjacent sequences: A349307 A349308 A349309 * A349311 A349312 A349313

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 14 2021

Status

approved