A361727 Diagonal of rational function 1/(1 - (1 + x*y) * (x^3 + y^3)).
1, 0, 0, 2, 4, 2, 6, 24, 36, 44, 126, 300, 470, 860, 2080, 4192, 7420, 15260, 33124, 64568, 124558, 259632, 535668, 1055460, 2118414, 4373412, 8872644, 17765396, 36138168, 73972404, 149793424, 303140552, 618565948, 1261454064, 2561056212, 5211145368
Offset: 0
Keywords
Links
- Winston de Greef, Table of n, a(n) for n = 0..3162
Programs
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PARI
a(n) = sum(k=0, n\3, binomial(2*k, k)*binomial(2*k, n-3*k));
Formula
G.f.: 1/sqrt(1 - 4 * x^3 * (1+x)^2).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(2*k,n-3*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: n*a(n) = 2*(2*n-3)*a(n-3) + 8*(n-2)*a(n-4) + 2*(2*n-5)*a(n-5).
a(n) ~ 1 / (sqrt((5 - 8*r^3 - 8*r^4)*Pi*n) * r^n), where r = 0.484163615233802299545617907511361266999078019358842974840776720... is the real root of the equation -1 + 4*r^3 + 8*r^4 + 4*r^5 = 0. (End)