A349532 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 4 * x).

1, 1, 7, 52, 407, 3329, 28232, 246552, 2204895, 20103027, 186223399, 1748009560, 16591329652, 158975004204, 1535725632552, 14940742412112, 146259921123407, 1439658075118967, 14240062489572485, 141469058343614452, 1410975387252602527, 14122900638031585153

Offset

0,3

Comments

In general, if k >= 0 and g.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - k*x), then a(n) ~ (4*k + 27)^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 4^(n+1)). - Vaclav Kotesovec, Nov 25 2021

Links

Table of n, a(n) for n=0..21.

Formula

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

a(n) = 4^(n-1)*F([4/3, 5/3, 1-n], [2, 5/2], -3^3/2^4), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021

a(n) ~ 43^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Nov 25 2021

Mathematica

nmax = 21; 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] = a[1] = 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, 21}]
Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 4^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]

Crossrefs

Cf. A001764, A025230, A307678, A349318, A349531.

Sequence in context: A370027 A147962 A329012 * A369147 A162233 A185623

Adjacent sequences: A349529 A349530 A349531 * A349533 A349534 A349535

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 21 2021

Status

approved