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%I #17 Aug 07 2023 08:31:49
%S 1,1,6,46,406,3901,39627,418592,4551672,50610692,572807157,6577068383,
%T 76426719408,897078662538,10620634999318,126676885170703,
%U 1520759193166329,18361269213121164,222814883564042704,2716125963857227904,33244557641365865109
%N G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).
%H Seiichi Manyama, <a href="/A349332/b349332.txt">Table of n, a(n) for n = 0..897</a>
%F a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
%F a(n) ~ 3381^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - _Vaclav Kotesovec_, Nov 15 2021
%F Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = (4405*n^4 - 10346*n^3 + 9575*n^2 - 4354*n + 840)*a(n-1) - 12*(n-2)*(1255*n^3 - 3957*n^2 + 4492*n - 1820)*a(n-2) + 2*(n-3)*(n-2)*(10655*n^2 - 32733*n + 26908)*a(n-3) - 4*(n-4)*(n-3)*(n-2)*(3445*n - 6986)*a(n-4) + 3381*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). - _Vaclav Kotesovec_, Nov 17 2021
%p a:= n-> coeff(series(RootOf(1+x*A^5/(1-x)-A, A), x, n+1), x, n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 15 2021
%t 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 Table[Sum[Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
%Y Cf. A002212, A307678, A349331, A349333, A349334, A349335.
%Y Cf. A002294, A346647 (partial sums), A349361, A364748.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 15 2021
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