A349312 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^6) / (1 - x).

1, 2, 14, 158, 2106, 30762, 476406, 7683926, 127692530, 2171184146, 37592376734, 660522703886, 11747865153962, 211093333172282, 3826315983647366, 69880933123237958, 1284661783610775010, 23753502514840942882, 441458929706855144494, 8242097867816771820926

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(6*k,k) / (5*k+1).

a(n) = F([(1+n)/5, (2+n)/5, (3+n)/5, (4+n)/5, 1+n/5, -n], [2/5, 3/5, 4/5, 1, 6/5], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021

a(n) ~ sqrt(1 + 5*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi) * (1-r)^(3/10) * n^(3/2) * r^(n + 1/5)), where r = 0.04941755525635041337247049893940451999923592381716... is the smallest real root of the equation 5^5 * (1-r)^6 = 6^6 * r. - Vaclav Kotesovec, Nov 15 2021

Mathematica

nmax = 19; A[_] = 0; Do[A[x_] = (1 + x A[x]^6)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

Crossrefs

Cf. A002295, A006318, A346626, A346648, A349310, A349311, A349313, A349314.

Sequence in context: A355114 A332693 A196791 * A218295 A354242 A268011

Adjacent sequences: A349309 A349310 A349311 * A349313 A349314 A349315

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 14 2021

Status

approved