A361728 Diagonal of rational function 1/(1 - (1 + x*y*z) * (x + y + z)).

1, 6, 108, 2238, 51126, 1234836, 30933846, 795124008, 20832161238, 553908550416, 14901620938668, 404737904238768, 11080360585597974, 305375448989901564, 8464333256181647028, 235772833122673888788, 6595763835075158604618

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

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

From Vaclav Kotesovec, Mar 22 2023: (Start)

Recurrence: (n-1)*n^2*a(n) = 2*(n-1)*(13*n^2 - 13*n + 3)*a(n-1) + 12*(9*n^3 - 27*n^2 + 23*n - 3)*a(n-2) + 18*(9*n^3 - 36*n^2 + 38*n - 3)*a(n-3) + 12*(9*n^3 - 45*n^2 + 56*n - 2)*a(n-4) + 3*n*(3*n - 11)*(3*n - 7)*a(n-5).

a(n) ~ c * d^n / n, where d = 29.8094342438507627973286122946283855557156321402886102401458498265933891... is the real root of the equation -27 - 81*d - 81*d^2 - 27*d^3 + d^4 = 0 and c = sqrt(3)/(2*Pi) = 0.27566444771089602475566324915648472069869324018332... (End)

Mathematica

Table[Sum[(3*k)!/k!^3 * Binomial[3*k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 22 2023 *)

Prog

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

Crossrefs

Cf. A361729, A361730.

Sequence in context: A099138 A230540 A295791 * A055532 A369532 A288148

Adjacent sequences: A361725 A361726 A361727 * A361729 A361730 A361731

Keyword

nonn

Author

Seiichi Manyama, Mar 22 2023

Status

approved