A349293 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)).

1, 2, 17, 249, 4345, 83285, 1694273, 35915349, 784691569, 17545398747, 399545961817, 9234298584921, 216053290499201, 5107287712887563, 121795876378121121, 2926604574330886897, 70788399943851406825, 1722188546498276868124, 42114624858397590035177

Offset: 0

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

• nonn

Comments

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

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

Programs

Mathematica

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

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

Formula

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

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

a(n) = 1 + Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 11 2025