A349591 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^6 * A(x)^8.

1, 3, 16, 160, 2216, 35110, 596016, 10573748, 193586424, 3629709697, 69342483276, 1344897261828, 26411276859800, 524117511080056, 10493756451964088, 211719733855698808, 4300202981875132408, 87854045612854431128, 1804215079309443709632

Offset: 0

Author

Ilya Gutkovskiy, Nov 22 2021

Keywords

• nonn

Comments

Second binomial transform of A007556.

Crossrefs

Cf. A007556, A064613, A346650, A346762, A349581, A349582, A349584, A349590.

Programs

Mathematica

nmax = 18; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^6 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[8 k, k] 2^(n - k)/(7 k + 1), {k, 0, n}], {n, 0, 18}]

PARI

a(n) = sum(k=0, n, binomial(n,k)*binomial(8*k,k)*2^(n-k)/(7*k+1)); \\ Michel Marcus, Nov 23 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(8*k,k) * 2^(n-k) / (7*k+1).

a(n) = 2^n*F([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -2^23/7^7), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021

a(n) ~ 2^(n - 67/2) * 9212151^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021