This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349293 #18 Jul 11 2025 09:05:51
%S A349293 1,2,17,249,4345,83285,1694273,35915349,784691569,17545398747,
%T A349293 399545961817,9234298584921,216053290499201,5107287712887563,
%U A349293 121795876378121121,2926604574330886897,70788399943851406825,1722188546498276868124,42114624858397590035177
%N A349293 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)).
%C A349293 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 A349293 Seiichi Manyama, <a href="/A349293/b349293.txt">Table of n, a(n) for n = 0..500</a>
%F A349293 a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).
%F A349293 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
%F A349293 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
%t A349293 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 A349293 Table[Sum[Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]
%o A349293 (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 A349293 Cf. A007317, A007556, A199475, A346650, A349289, A349290, A349291, A349292.
%K A349293 nonn
%O A349293 0,2
%A A349293 _Ilya Gutkovskiy_, Nov 13 2021