A346762 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x) * A(x)^3.

1, 3, 11, 50, 271, 1655, 10900, 75388, 539295, 3954593, 29557251, 224308078, 1723659436, 13384272660, 104855628776, 827760536528, 6578127170319, 52581460222645, 422478996770305, 3410174204693310, 27640220748529799, 224866485110361767, 1835589569664256976

Offset: 0

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

• nonn

Comments

Second binomial transform of A001764.

Crossrefs

Cf. A001764, A064613, A188687, A346763.

Programs

Mathematica

nmax = 22; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x) A[x]^3 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
Table[2^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -27/8], {n, 0, 22}]

Formula

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

a(n) ~ 35^(n + 3/2) / (81 * sqrt(Pi) * n^(3/2) * 4^(n+1)). - Vaclav Kotesovec, Nov 26 2021