A349516 G.f. A(x) satisfies: A(x) = (1 + 3 * x * A(x)^3) / (1 - x).

1, 4, 40, 544, 8512, 144448, 2584960, 48026368, 917535232, 17911696384, 355725727744, 7164414312448, 145983839272960, 3003998986682368, 62337412584669184, 1303045468017786880, 27411525832634269696, 579884892273731436544, 12328565505725394583552

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

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

a(n) ~ sqrt(13 + 7*3^(1/3) + 5*3^(2/3)) / (12 * sqrt(Pi) * n^(3/2) * (1 + 3^(4/3)/2 - 3^(5/3)/2)^n). - Vaclav Kotesovec, Nov 21 2021

Mathematica

nmax = 18; A[_] = 0; Do[A[x_] = (1 + 3 x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + 3 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, 18}]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 3^k/(2 k + 1), {k, 0, n}], {n, 0, 18}]

Prog

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

Crossrefs

Cf. A001764, A103211, A346626, A348912, A349517.

Sequence in context: A371676 A075878 A092812 * A290575 A196867 A276362

Adjacent sequences: A349513 A349514 A349515 * A349517 A349518 A349519

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 20 2021

Status

approved