A346627 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * A(x)^3.

1, 0, 1, 2, 7, 23, 82, 300, 1129, 4334, 16914, 66899, 267586, 1080516, 4398850, 18035084, 74402361, 308624282, 1286428765, 5385578256, 22635057148, 95471113565, 403983783772, 1714494024947, 7295949019114, 31124885587680, 133085594104222, 570266646942488

Offset: 0

Author

Ilya Gutkovskiy, Jul 25 2021

Keywords

• nonn

Comments

Inverse binomial transform of A200753.

Links

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

Crossrefs

Cf. A001764, A005043, A199475, A200753, A346628, A349299, A349300, A349301, A349302, A349303.

Programs

Mathematica

nmax = 27; A[] = 0; Do[A[x] = 1/(1 + x) + x A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 27}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 27}]

Formula

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

a(n) = (-1)^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).

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

a(n) ~ sqrt(198 + 38*sqrt(33)) * (19 + 3*sqrt(33))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 3)). - Vaclav Kotesovec, Jul 30 2021