A349533 G.f. A(x) satisfies A(x) = 1 / ((1 - 2 * x) * (1 - x * A(x)^2)).

1, 3, 13, 74, 499, 3719, 29494, 243888, 2078431, 18122369, 160885449, 1449268478, 13213370392, 121696581804, 1130565483472, 10581614352704, 99685591788687, 944490400760597, 8994266558594671, 86040075341770806, 826423263373253923, 7967095415955791687

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..989

Formula

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

a(n) = 2^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^3/2^5), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021

a(n) ~ 177^(1/4) * (43 + 3*sqrt(177))^(n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 22 2021

Mathematica

nmax = 21; A[_] = 0; Do[A[x_] = 1/((1 - 2 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 2^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, 21}]
Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]

Crossrefs

Cf. A001764, A064613, A199475, A349253, A349534, A349535.

Sequence in context: A190878 A156154 A334785 * A020094 A189886 A343787

Adjacent sequences: A349530 A349531 A349532 * A349534 A349535 A349536

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 21 2021

Status

approved