A346648 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) / (5*k + 1).

1, 2, 9, 73, 751, 8587, 104425, 1323952, 17303503, 231455104, 3153167249, 43597546197, 610232050453, 8629733401556, 123114479858631, 1769728635257503, 25607523627970183, 372688563309335806, 5451995469296025115, 80122698147986922194, 1182341393088427774071

Offset

0,2

Comments

Binomial transform of A002295.

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)^4 * A(x)^6.

G.f.: Sum_{k>=0} ( binomial(6*k,k) / (5*k + 1) ) * x^k / (1 - x)^(k+1).

a(n) ~ 49781^(n + 3/2) / (3359232 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Mathematica

Table[Sum[Binomial[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6, -n}, {2/5, 3/5, 4/5, 1, 6/5}, -46656/3125], {n, 0, 20}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(6*k, k)/(5*k + 1)); \\ Michel Marcus, Jul 26 2021

Crossrefs

Cf. A002295, A007317, A188687, A226910, A346646, A346647, A346649, A346650.

Sequence in context: A108995 A184358 A132843 * A336955 A084873 A332722

Adjacent sequences: A346645 A346646 A346647 * A346649 A346650 A346651

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 26 2021

Status

approved