A346763 G.f. A(x) satisfies: A(x) = 1 / (1 - 3*x) + x * (1 - 3*x) * A(x)^3.

1, 4, 18, 93, 550, 3636, 26079, 197931, 1562382, 12685116, 105187512, 886700898, 7574331987, 65413265014, 570155069547, 5008957733472, 44306834969838, 394269180748272, 3527034255411864, 31700659283908242, 286124960854479888, 2592334353741781752, 23567790327842864046

Offset: 0

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

• nonn

Comments

Third binomial transform of A001764.

Crossrefs

Cf. A001764, A104455, A188687, A346762.

Programs

Mathematica

nmax = 22; A[] = 0; Do[A[x] = 1/(1 - 3 x) + x (1 - 3 x) A[x]^3 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
Table[3^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -9/4], {n, 0, 22}]

Formula

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

a(n) ~ 3^(n - 5/2) * 13^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*(n+1))). - Vaclav Kotesovec, Nov 26 2021