A346672 - OEIS

%I #14 Jul 30 2021 10:19:51

%S 1,2,10,102,1342,19620,305004,4943352,82595376,1412486081,24602515801,

%T 434935956337,7783978950825,140752989839105,2567623696254905,

%U 47195200645619009,873239636055018809,16251426606785706209,304007720310330530081,5713101394865420846381

%N a(n) = Sum_{k=0..n} binomial(8*k,k) / (7*k + 1).

%C Partial sums of A007556.

%C In general, for m > 1, Sum_{k=0..n} binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m - (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021

%H Seiichi Manyama, <a href="/A346672/b346672.txt">Table of n, a(n) for n = 0..768</a>

%F G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^7 * A(x)^8.

%F a(n) ~ 2^(24*n + 25) / (15953673 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021

%t Table[Sum[Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]

%t nmax = 19; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%o (PARI) a(n) = sum(k=0, n, binomial(8*k, k)/(7*k+1)); \\ _Michel Marcus_, Jul 28 2021

%Y Cf. A007556, A014137, A104859, A345367, A345368, A346065, A346650, A346671.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jul 28 2021