A349313 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^7) / (1 - x).

1, 2, 16, 212, 3320, 57024, 1038928, 19718512, 385668448, 7718866880, 157326086656, 3254310606208, 68142850580480, 1441588339943168, 30765576147680000, 661561298256228096, 14319744815795062272, 311756656998135770112, 6822215641015820419072

Offset

0,2

Links

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

Formula

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

a(n) = F([(1+n)/6, (2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021

a(n) ~ sqrt(1 + 6*r) / (2 * 7^(2/3) * sqrt(3*Pi) * (1-r)^(1/3) * n^(3/2) * r^(n + 1/6)), where r = 0.04196526794785323647696104132939153750367778616407409162... is the real root of the equation 6^6 * (1-r)^7 = 7^7 * r. - Vaclav Kotesovec, Nov 15 2021

Mathematica

nmax = 18; A[_] = 0; Do[A[x_] = (1 + x A[x]^7)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

Crossrefs

Cf. A002296, A006318, A346626, A346649, A349310, A349311, A349312, A349314.

Sequence in context: A188600 A206930 A206865 * A364399 A365568 A360939

Adjacent sequences: A349310 A349311 A349312 * A349314 A349315 A349316

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 14 2021

Status

approved