A346666 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(6*k,k) / (5*k + 1).

1, 0, 5, 35, 335, 3405, 36601, 408630, 4693535, 55105970, 658390845, 7979041735, 97847884981, 1211946011450, 15139726594915, 190526268260405, 2413170608875655, 30738613968350640, 393519782671609951, 5060600804169151680, 65342131689498876095, 846781225288921612940

Offset

0,3

Comments

Inverse 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) ~ 43531^(n + 3/2) / (3359232 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Mathematica

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; 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 = 21; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[(-1)^n 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, 21}]

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(6*k, k)/(5*k + 1)); \\ Michel Marcus, Jul 28 2021

Crossrefs

Cf. A002295, A005043, A346628, A346648, A346664, A346665, A346667, A346668.

Sequence in context: A305306 A113342 A262248 * A369724 A360611 A201367

Adjacent sequences: A346663 A346664 A346665 * A346667 A346668 A346669

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 27 2021

Status

approved