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A336540 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (2 + A(x)).

%I A336540 #30 Aug 09 2025 20:55:38
%S A336540 1,3,39,705,14799,338430,8181597,205655304,5320853535,140761481673,
%T A336540 3790170529806,103531954949526,2861975199328581,79913364673955880,
%U A336540 2250605304332901048,63855671824327590480,1823518942311678061503,52371578117120237562459,1511737669565948867948805
%N A336540 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (2 + A(x)).
%H A336540 Seiichi Manyama, <a href="/A336540/b336540.txt">Table of n, a(n) for n = 0..671</a>
%F A336540 a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(4*n,k-1) for n > 0.
%F A336540 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).
%F A336540 a(n) = (1/(4*n+1)) * Sum_{k=0..n} 2^k * binomial(4*n+1,k) * binomial(5*n-k,n-k).
%F A336540 a(n) ~ sqrt(59601 + 205733*sqrt(3/43)) * (7781 + 731*sqrt(129))^(n - 1/2) / (sqrt(Pi) * n^(3/2) * 2^(9*n + 7/2)). - _Vaclav Kotesovec_, Jul 31 2021
%F A336540 a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0. - _Seiichi Manyama_, Aug 10 2023
%F A336540 a(n) = 2^n*binomial(1+4*n, n)*hypergeom([-n, 1+4*n], [2+3*n], -1/2)/(1 + 4*n). - _Stefano Spezia_, Aug 09 2025
%t A336540 a[n_] := Sum[2^(n-k) * Binomial[n, k] * Binomial[4*n + k + 1, n]/(4*n + k + 1), {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Jul 28 2020 *)
%o A336540 (PARI) a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^4*(2+A)); polcoeff(A, n);
%o A336540 (PARI) a(n) = if(n==0, 1, sum(k=1, n, 3^k*binomial(n, k)*binomial(4*n, k-1)/n));
%o A336540 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1)); \\ _Seiichi Manyama_, Jul 28 2020
%o A336540 (PARI) a(n) = sum(k=0, n, 2^k*binomial(4*n+1, k)*binomial(5*n-k, n-k))/(4*n+1); \\ _Seiichi Manyama_, Jul 28 2020
%Y A336540 Column k=4 of A336575.
%K A336540 nonn
%O A336540 0,2
%A A336540 _Seiichi Manyama_, Jul 25 2020