A349534 G.f. A(x) satisfies A(x) = 1 / ((1 - 3 * x) * (1 - x * A(x)^2)).

1, 4, 21, 138, 1063, 9075, 82770, 789204, 7766721, 78267306, 803447526, 8371413999, 88300495746, 941004684748, 10116276976218, 109578418285452, 1194764348642313, 13102287157827918, 144422108994233625, 1599198859915070640, 17780781456147340764

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..931

Formula

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

a(n) = 3^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^2/2^4), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021

a(n) ~ sqrt(5/Pi) * 3^(n-1) * 4^n / n^(3/2). - Vaclav Kotesovec, Nov 22 2021

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = 1/((1 - 3 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 3^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, 20}]
Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 20}]

Crossrefs

Cf. A001764, A104455, A199475, A349254, A349533, A349535.

Sequence in context: A369784 A121124 A180399 * A222058 A265174 A087761

Adjacent sequences: A349531 A349532 A349533 * A349535 A349536 A349537

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 21 2021

Status

approved