A361673 Constant term in the expansion of (1 + x*y + y*z + z*x + 1/(x*y*z))^n.

1, 1, 1, 1, 1, 61, 361, 1261, 3361, 7561, 34021, 235621, 1294921, 5482621, 19039021, 65345281, 286147681, 1511480881, 7688794681, 34337600281, 138221512741, 554603041441, 2454508134541, 11874549049441, 57412094595241, 261925516443361, 1134301869703861

Offset

0,6

Comments

Also constant term in the expansion of (1 + x^2 + y^2 + z^2 + 1/(x*y*z))^n.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = n! * Sum_{k=0..floor(n/5)} 1/(k!^3 * (2*k)! * (n-5*k)!) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * A001460(k).

From Vaclav Kotesovec, Mar 22 2023: (Start)

Recurrence: 2*n^3*(2*n - 5)*a(n) = 2*(10*n^4 - 40*n^3 + 50*n^2 - 30*n + 7)*a(n-1) - 10*(n-1)*(4*n^3 - 18*n^2 + 26*n - 13)*a(n-2) + 40*(n-2)^3*(n-1)*a(n-3) - 10*(n-3)*(n-2)*(n-1)*(2*n - 5)*a(n-4) + 3129*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).

a(n) ~ sqrt(c) * (1 + 5/2^(2/5))^n / (Pi^(3/2) * n^(3/2)), where c = 3.154712586460560795509193778252140601572145506226776094640234924884123818... is the real root of the equation -30634915689 + 95407210000*c - 127160000000*c^2 + 79846400000*c^3 - 25600000000*c^4 + 3276800000*c^5 = 0. (End)

Mathematica

Table[n! * Sum[1/(k!^3 * (2*k)! * (n-5*k)!), {k, 0, n/5}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 22 2023 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\5, 1/(k!^3*(2*k)!*(n-5*k)!));

Crossrefs

Cf. A344560, A361675.

Cf. A361637, A361658.

Cf. A001460.

Sequence in context: A142953 A245763 A130117 * A373525 A357965 A353225

Adjacent sequences: A361670 A361671 A361672 * A361674 A361675 A361676

Keyword

nonn

Author

Seiichi Manyama, Mar 20 2023

Status

approved