A346649 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k,k) / (6*k + 1).

1, 2, 10, 95, 1146, 15343, 218407, 3241316, 49588850, 776483636, 12383420161, 200444399493, 3284531747403, 54378741581471, 908238222519566, 15284835717461020, 258933935458506210, 4412025177612412048, 75564998345532498844, 1300158755391113561288

Offset

0,2

Comments

Binomial transform of A002296.

Links

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

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^7.

G.f.: Sum_{k>=0} ( binomial(7*k,k) / (6*k + 1) ) * x^k / (1 - x)^(k+1).

a(n) ~ 870199^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Mathematica

Table[Sum[Binomial[n, k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 19; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n}, {1/3, 1/2, 2/3, 5/6, 1, 7/6}, -823543/46656], {n, 0, 19}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(7*k, k)/(6*k + 1)); \\ Michel Marcus, Jul 26 2021

Crossrefs

Cf. A002296, A007317, A188687, A346646, A346647, A346648, A346650.

Sequence in context: A182173 A362643 A160940 * A355780 A193290 A193435

Adjacent sequences: A346646 A346647 A346648 * A346650 A346651 A346652

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 26 2021

Status

approved