OFFSET
0,2
COMMENTS
Second binomial transform of A002296.
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 22 2021
STATUS
approved