A349299 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^3)).
1, 0, 1, 3, 13, 57, 265, 1273, 6281, 31634, 161985, 840795, 4413937, 23395376, 125028541, 672958929, 3644868105, 19850452482, 108639736105, 597190462341, 3295737021241, 18253432801480, 101425665527825, 565249069709027, 3158734991846065
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
nmax = 24; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x]^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Table[Sum[(-1)^(n - k) Binomial[n + 2 k, 3 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 24}]
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PARI
a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1)); \\ Michel Marcus, Nov 14 2021
Formula
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1).
a(n) = (-1)^n*F([1/4, 1/2, 3/4, (1+n)/2, (2+n)/2, -n], [1/3, 2/3, 2/3, 1, 4/3], 2^10/3^6), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 - 2*r) / (4 * 2^(1/6) * sqrt(3*Pi*(1+r)) * n^(3/2) * r^(n + 1/3)), where r = 0.16809738261179529189597734361984743421471587505782537522127218... is the root of the equation 4^4 * r = 3^3 * (1+r)^3. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^3), where F(x) = Sum_{n >= 0} A002293(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^4. (End)