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A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

%I A349335 #22 Nov 23 2024 09:10:30
%S A349335 1,1,9,109,1541,23823,390135,6651051,116798643,2098313686,38382509118,
%T A349335 712447023590,13385500614902,254065657922154,4864482597112186,
%U A349335 93840443376075810,1822169236520766546,35586928273002974487,698572561837366684479,13775697096997873764647
%N A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).
%C A349335 In general, for m > 1, Sum_{k=0..n} binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (1 + m^m/(m-1)^(m-1))^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - _Vaclav Kotesovec_, Nov 15 2021
%H A349335 Seiichi Manyama, <a href="/A349335/b349335.txt">Table of n, a(n) for n = 0..756</a>
%F A349335 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).
%F A349335 a(n) ~ 17600759^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - _Vaclav Kotesovec_, Nov 15 2021
%p A349335 a:= n-> coeff(series(RootOf(1+x*A^8/(1-x)-A, A), x, n+1), x, n):
%p A349335 seq(a(n), n=0..19);  # _Alois P. Heinz_, Nov 15 2021
%t A349335 nmax = 19; A[_] = 0; Do[A[x_] = 1 + x A[x]^8/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349335 Table[Sum[Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
%o A349335 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
%o A349335 A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^8, k)) )); A[n+1]}
%o A349335 for(n=0, 30, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Nov 23 2024, after _Paul D. Hanna_
%Y A349335 Cf. A002212, A307678, A349331, A349332, A349333, A349334.
%Y A349335 Cf. A007556, A346650 (partial sums), A349364.
%K A349335 nonn
%O A349335 0,3
%A A349335 _Ilya Gutkovskiy_, Nov 15 2021