A336572 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (1 + 2 * A(x)).
1, 3, 42, 822, 18708, 464115, 12175368, 332156784, 9328004700, 267870927324, 7829893576878, 232189300430454, 6968123350684692, 211232335919261178, 6458598626291716128, 198949096401788859636, 6168233789851179030684, 192334850789654814053700, 6027727888877572168027368
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..655
Programs
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Mathematica
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 *) -
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);
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PARI
a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1));
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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
Formula
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).
a(n) = (1/(4*n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(5*n-k,n-k).
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
From Seiichi Manyama, Aug 10 2023: (Start)
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.
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)
a(n) = binomial(1+4*n, n)*hypergeom([-n, 1+4*n], [2+3*n], -2)/(1 + 4*n). - Stefano Spezia, Aug 09 2025