A355293 Expansion of e.g.f. 1 / (1 - x - x^2/2 - x^3/3).

1, 1, 3, 14, 82, 610, 5450, 56700, 674520, 9027480, 134236200, 2195701200, 39180094800, 757389032400, 15767305554000, 351689317980000, 8367381470448000, 211518767796336000, 5661504152255952000, 159954273475764768000, 4757034049019572320000, 148547713504322452320000, 4859583724723970642400000

Offset

0,3

Comments

a(n) is the number of ways an n-person race can end if two-way and three way ties are allowed (see Bona at pp. 401-402). - Stefano Spezia, Apr 06 2026

References

Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, pp. 401-402.

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = n * a(n-1) + n * (n-1) * a(n-2) / 2 + n * (n-1) * (n-2) * a(n-3) / 3.

Mathematica

nmax = 22; CoefficientList[Series[1/(1 - x - x^2/2 - x^3/3), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[2] = 3; a[n_] := a[n] = n a[n - 1] + n (n - 1) a[n - 2]/2 + n (n - 1) (n - 2) a[n - 3]/3; Table[a[n], {n, 0, 22}]

Crossrefs

Cf. A007840, A057693, A080599, A189886, A355294.

Sequence in context: A020104 A220894 A103467 * A215661 A224776 A220910

Adjacent sequences: A355290 A355291 A355292 * A355294 A355295 A355296

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 27 2022

Status

approved