OFFSET
0,2
COMMENTS
In general, for k > 1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial(k*j,j) / ((k-1)*j+1) ~ (1-r)^(1/(k-1) - 1/2) * sqrt(1 + (k-1)*r) / (sqrt(2*Pi*(k-1)) * k^(1/(k-1) + 1/2) * n^(3/2) * r^(n + 1/(k-1))), where r is the smallest real root of the equation (k-1)^(k-1) * (1-r)^k = k^k * r. - Vaclav Kotesovec, Nov 15 2021
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{k=0..n} binomial(n+7*k,8*k) * binomial(8*k,k) / (7*k+1).
a(n) = F([(1+n)/7, (2+n)/7, (3+n)/7, (4+n)/7, (5+n)/7, (6+n)/7, 1+n/7, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 7*r) / (2^(17/7) * sqrt(7*Pi) * (1-r)^(5/14) * n^(3/2) * r^(n + 1/7)), where r = 0.036466941615119756839260438459647497790132092200414533994... is the smallest real root of the equation 7^7 * (1-r)^8 = 8^8 * r. - Vaclav Kotesovec, Nov 15 2021
MATHEMATICA
nmax = 18; A[_] = 0; Do[A[x_] = (1 + x A[x]^8)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 7 k, 8 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 14 2021
STATUS
approved