A349515 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - 4 * x).

1, 5, 35, 320, 3415, 39805, 490660, 6288120, 82935615, 1118324655, 15346920635, 213637539620, 3009391426340, 42817011909180, 614411343795960, 8881874095390320, 129224763346019215, 1890813939312392755, 27805864640943573385, 410748152876389349720

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

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

a(n) ~ 2^(4*n + 1/2) / (sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 21 2021

Mathematica

nmax = 19; A[_] = 0; Do[A[x_] = (1 + x A[x]^3)/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 4 a[n - 1] + 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, 19}]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 4^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 19}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+2*k, 3*k) * binomial(3*k, k) * 4^(n-k) / (2*k+1)) \\ Andrew Howroyd, Nov 20 2021

Crossrefs

Cf. A001764, A082301, A346626, A348793, A349514.

Sequence in context: A051577 A124564 A260952 * A362356 A307679 A305306

Adjacent sequences: A349512 A349513 A349514 * A349516 A349517 A349518

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 20 2021

Status

approved