A349333 - OEIS

%I #23 Nov 23 2024 09:09:48

%S 1,1,7,64,678,7836,95838,1219527,15979551,214151601,2921712145,

%T 40444378948,566634504256,8019501351103,114484746457075,

%U 1646614155398872,23837794992712680,347081039681365623,5079306905986689309,74670702678690897079,1102218694940440851877

%N G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

%H Seiichi Manyama, <a href="/A349333/b349333.txt">Table of n, a(n) for n = 0..836</a>

%F a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(6*k,k) / (5*k+1).

%F a(n) ~ 49781^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - _Vaclav Kotesovec_, Nov 15 2021

%p a:= n-> coeff(series(RootOf(1+x*A^6/(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]^6/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t Table[Sum[Binomial[n - 1, k - 1] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

%o (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);

%o A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^6, k)) )); A[n+1]}

%o for(n=0, 30, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Nov 23 2024, after _Paul D. Hanna_

%Y Cf. A002212, A307678, A349331, A349332, A349334, A349335.

%Y Cf. A002295, A346648 (partial sums), A349362.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 15 2021