A361729 Diagonal of rational function 1/(1 - (1 + x*y*z) * (x^2 + y^2 + z^2)).

1, 0, 6, 18, 108, 546, 3030, 16920, 96480, 557460, 3255426, 19186020, 113905386, 680583708, 4088506428, 24677473884, 149564145060, 909784736388, 5552109174084, 33981183515664, 208523253915306, 1282621025382840, 7906367632595328, 48832556909752044

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1230

Formula

a(n) = Sum_{k=0..floor(n/2)} (3*k)!/k!^3 * binomial(3*k,n-2*k).

From Vaclav Kotesovec, Mar 22 2023: (Start)

Recurrence: (n-1)*n^2*a(n) = -(n-1)^2*n*a(n-1) + 3*(n-1)*(3*n - 4)*(3*n - 2)*a(n-2) + 12*(9*n^3 - 36*n^2 + 41*n - 9)*a(n-3) + 18*(3*n - 8)*(3*n^2 - 7*n + 1)*a(n-4) + 12*(9*n^3 - 54*n^2 + 80*n - 5)*a(n-5) + 3*n*(3*n - 13)*(3*n - 8)*a(n-6).

a(n) ~ c * d^n / n, where d = 6.45021022459140188868150633620495776554217848977385402261531271... is the real root of the equation -27 - 81*d - 81*d^2 - 27*d^3 + d^5 = 0 and c = sqrt(3)/(2*Pi) = 0.275664447710896024755663249156484720698693240183320326399... (End)

Prog

(PARI) a(n) = sum(k=0, n\2, (3*k)!/k!^3*binomial(3*k, n-2*k));

Crossrefs

Cf. A361728, A361730.

Sequence in context: A181038 A222857 A367664 * A108735 A143556 A007126

Adjacent sequences: A361726 A361727 A361728 * A361730 A361731 A361732

Keyword

nonn

Author

Seiichi Manyama, Mar 22 2023

Status

approved