A349581 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^2 * A(x)^4.

1, 3, 12, 66, 460, 3681, 31848, 289176, 2714044, 26103468, 255876048, 2546717454, 25666830724, 261407935366, 2686191839232, 27815564456544, 289960011573212, 3040424427011492, 32046741183678288, 339345854532800136, 3608307717155678256, 38511520730570169033

Offset

0,2

Comments

Second binomial transform of A002293.

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) * 2^(n-k) / (3*k+1).

a(n) = 2^n*F([1/4, 1/2, 3/4, -n], [2/3, 1, 4/3], -2^7/3^3), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021

a(n) ~ 2^(n - 10) * 155^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021

Mathematica

nmax = 21; A[_] = 0; Do[A[x_] = 1/(1 - 2 x) + x (1 - 2 x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[4 k, k] 2^(n - k)/(3 k + 1), {k, 0, n}], {n, 0, 21}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(4*k, k)*2^(n-k)/(3*k+1)); \\ Michel Marcus, Nov 23 2021

Crossrefs

Cf. A002293, A064613, A346646, A346762, A349582, A349584, A349590, A349591.

Sequence in context: A293302 A248871 A080599 * A120575 A009362 A123227

Adjacent sequences: A349578 A349579 A349580 * A349582 A349583 A349584

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 22 2021

Status

approved