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A349361 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^5 / (1 + x).

%I A349361 #17 Nov 20 2021 07:20:55
%S A349361 1,1,4,26,194,1581,13625,122120,1126780,10631460,102104845,994855179,
%T A349361 9809872626,97710157154,981636609906,9935473707279,101214412755647,
%U A349361 1036991125300748,10678412226507032,110459290208905008,1147261657267290037
%N A349361 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^5 / (1 + x).
%H A349361 Seiichi Manyama, <a href="/A349361/b349361.txt">Table of n, a(n) for n = 0..500</a>
%F A349361 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
%F A349361 a(n) = (-1)^(n+1)*F([6/5, 7/5, 8/5, 9/5, 1-n], [3/2, 7/4, 2, 9/4], 5^5/2^8), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 15 2021
%F A349361 From _Vaclav Kotesovec_, Nov 17 2021: (Start)
%F A349361 a(n) ~ 2869^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)).
%F A349361 Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = 3*(615*n^4 - 718*n^3 - 275*n^2 + 618*n - 200)*a(n-1) + 4*(n-2)*(2485*n^3 - 6879*n^2 + 6524*n - 2040)*a(n-2) + 2*(n-3)*(n-2)*(8095*n^2 - 23517*n + 18092)*a(n-3) + 12*(n-4)*(n-3)*(n-2)*(935*n - 1838)*a(n-4) + 2869*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). (End)
%p A349361 a:= n-> coeff(series(RootOf(1+x*A^5/(1+x)-A, A), x, n+1), x, n):
%p A349361 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 15 2021
%t A349361 nmax = 20; A[_] = 0; Do[A[x_] = 1 + x A[x]^5/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349361 Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
%Y A349361 Cf. A001006, A002294, A127897, A317133, A345368 (binomial transform), A346665, A349332, A349362, A349363, A349364.
%K A349361 nonn
%O A349361 0,3
%A A349361 _Ilya Gutkovskiy_, Nov 15 2021