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A336572 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (1 + 2 * A(x)).

%I A336572 #36 Aug 09 2025 20:54:40
%S A336572 1,3,42,822,18708,464115,12175368,332156784,9328004700,267870927324,
%T A336572 7829893576878,232189300430454,6968123350684692,211232335919261178,
%U A336572 6458598626291716128,198949096401788859636,6168233789851179030684,192334850789654814053700,6027727888877572168027368
%N A336572 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (1 +  2 * A(x)).
%H A336572 Seiichi Manyama, <a href="/A336572/b336572.txt">Table of n, a(n) for n = 0..655</a>
%F A336572 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).
%F A336572 a(n) = (1/(4*n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(5*n-k,n-k).
%F A336572 a(n) ~ sqrt(95781603 + 7199237*sqrt(177))*(69845 + 5251*sqrt(177))^(n - 1/2) / (sqrt(59*Pi) * n^(3/2) * 2^(12*n + 9/2)). - _Vaclav Kotesovec_, Jul 31 2021
%F A336572 From _Seiichi Manyama_, Aug 10 2023: (Start)
%F A336572 a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 3^(n-k) * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0.
%F A336572 a(n) = (1/n) * Sum_{k=1..n} 3^k * 2^(n-k) * binomial(n,k) * binomial(4*n,k-1) for n > 0. (End)
%F A336572 a(n) = binomial(1+4*n, n)*hypergeom([-n, 1+4*n], [2+3*n], -2)/(1 + 4*n). - _Stefano Spezia_, Aug 09 2025
%t A336572 a[n_] := Sum[2^k * Binomial[n, k] * Binomial[4*n + k + 1, n]/(4*n + k + 1), {k, 0, n}];  Array[a, 19, 0] (* _Amiram Eldar_, Jul 27 2020 *)
%o A336572 (PARI) a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^4*(1+2*A)); polcoeff(A, n);
%o A336572 (PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1));
%o A336572 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k)*binomial(5*n-k, n-k))/(4*n+1); \\ _Seiichi Manyama_, Jul 26 2020
%Y A336572 Column k=4 of A336574.
%Y A336572 Cf. A243667, A336540.
%K A336572 nonn
%O A336572 0,2
%A A336572 _Seiichi Manyama_, Jul 25 2020