The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #9 Apr 06 2026 09:54:56
%S 1,1,3,14,82,610,5450,56700,674520,9027480,134236200,2195701200,
%T 39180094800,757389032400,15767305554000,351689317980000,
%U 8367381470448000,211518767796336000,5661504152255952000,159954273475764768000,4757034049019572320000,148547713504322452320000,4859583724723970642400000
%N Expansion of e.g.f. 1 / (1 - x - x^2/2 - x^3/3).
%C 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
%D Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, pp. 401-402.
%F a(n) = n * a(n-1) + n * (n-1) * a(n-2) / 2 + n * (n-1) * (n-2) * a(n-3) / 3.
%t nmax = 22; CoefficientList[Series[1/(1 - x - x^2/2 - x^3/3), {x, 0, nmax}], x] Range[0, nmax]!
%t 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}]
%Y Cf. A007840, A057693, A080599, A189886, A355294.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 27 2022