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A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

%I A349334 #22 Nov 23 2024 09:10:10
%S A349334 1,1,8,85,1051,14197,203064,3022909,46347534,726894786,11606936525,
%T A349334 188060979332,3084087347910,51094209834068,853859480938095,
%U A349334 14376597494941454,243649099741045190,4153091242153905838,71152973167920086796,1224593757045581062444
%N A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).
%H A349334 Seiichi Manyama, <a href="/A349334/b349334.txt">Table of n, a(n) for n = 0..791</a>
%F A349334 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(7*k,k) / (6*k+1).
%F A349334 a(n) ~ 870199^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - _Vaclav Kotesovec_, Nov 15 2021
%p A349334 a:= n-> coeff(series(RootOf(1+x*A^7/(1-x)-A, A), x, n+1), x, n):
%p A349334 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 15 2021
%t A349334 nmax = 19; A[_] = 0; Do[A[x_] = 1 + x A[x]^7/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349334 Table[Sum[Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
%o A349334 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
%o A349334 A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^7, k)) )); A[n+1]}
%o A349334 for(n=0, 30, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Nov 23 2024, after _Paul D. Hanna_
%Y A349334 Cf. A002212, A307678, A349331, A349332, A349333, A349335.
%Y A349334 Cf. A002296, A346649 (partial sums), A349363.
%K A349334 nonn
%O A349334 0,3
%A A349334 _Ilya Gutkovskiy_, Nov 15 2021