A349590 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^5 * A(x)^7.

1, 3, 15, 132, 1595, 22134, 329718, 5136028, 82579819, 1359902823, 22818697128, 388728802702, 6705324823466, 116878939752376, 2055505806198352, 36427660285955808, 649894104351874395, 11662729497015257677, 210383830525447606431, 3812719304673511150854

Offset

0,2

Comments

Second binomial transform of A002296.

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k,k) * 2^(n-k) / (6*k+1).

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

a(n) ~ 916855^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 3^(6*n + 3/2) * 4^(3*n + 1)). - Vaclav Kotesovec, Nov 26 2021

Mathematica

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

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(7*k, k)*2^(n-k)/(6*k+1)); \\ Michel Marcus, Nov 23 2021

Crossrefs

Cf. A002296, A064613, A346649, A346762, A349581, A349582, A349584, A349591.

Sequence in context: A264795 A222390 A281186 * A108210 A006717 A222263

Adjacent sequences: A349587 A349588 A349589 * A349591 A349592 A349593

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 22 2021

Status

approved