A346628 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x) * A(x)^3.

1, 0, 2, 5, 22, 92, 415, 1927, 9198, 44804, 221880, 1113730, 5653747, 28975962, 149725355, 779178092, 4080167790, 21483383992, 113670233848, 604070682354, 3222823434608, 17255628041720, 92689459311470, 499359484166994, 2697571066055611, 14608820993453132

Offset: 0

Author

Ilya Gutkovskiy, Jul 25 2021

Keywords

• nonn

Comments

Inverse binomial transform of A001764.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001764, A005043, A188678, A188687, A346627, A346664, A346665, A346666, A346667, A346668.

Programs

Mathematica

nmax = 25; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x) A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 25; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 25}]

Formula

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 + x)^(k+1).

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

a(n) ~ 23^(n + 3/2) / (81 * sqrt(Pi) * n^(3/2) * 2^(2*n+2)). - Vaclav Kotesovec, Jul 30 2021

D-finite with recurrence +2*n*(2*n+1)*a(n) -(15*n-4)*(n-1)*a(n-1) -2*(n-1)*(21*n-22)*a(n-2) -23*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Aug 05 2021