A349293 - OEIS

%I #15 Nov 14 2021 05:06:43

%S 1,2,17,249,4345,83285,1694273,35915349,784691569,17545398747,

%T 399545961817,9234298584921,216053290499201,5107287712887563,

%U 121795876378121121,2926604574330886897,70788399943851406825,1722188546498276868124,42114624858397590035177

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

%C In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - _Vaclav Kotesovec_, Nov 14 2021

%H Seiichi Manyama, <a href="/A349293/b349293.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).

%F a(n) ~ sqrt(1 + 6*r) / (2^(17/7) * sqrt(7*Pi*(1-r)) * n^(3/2) * r^(n + 1/7)), where r = 0.0375502499742240443056934699070050852345109331376051496159609551... is the real root of the equation 8^8 * r = 7^7 * (1-r)^7. - _Vaclav Kotesovec_, Nov 14 2021

%t nmax = 18; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^7)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t Table[Sum[Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]

%o (PARI) a(n) = sum(k=0, n, binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1)); \\ _Michel Marcus_, Nov 14 2021

%Y Cf. A007317, A007556, A199475, A346650, A349289, A349290, A349291, A349292.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 13 2021