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%I #32 Feb 17 2022 13:20:54
%S 1,1,2,5,1,17,1,6,74,11,15,20,394,56,60,120,90,2484,407,525,490,630,
%T 504,18108,3235,4725,2240,4620,4032,3360,149904,29143,40509,27440,
%U 26460,33264,30240,25920,1389456,291394,398790,319760,163800,302400,277200,259200,226800
%N Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).
%C If the permutation has no third cycle, then its third-longest cycle is defined to have length 0.
%H Alois P. Heinz, <a href="/A350016/b350016.txt">Rows n = 0..142, flattened</a>
%H Steven Finch, <a href="http://arxiv.org/abs/2202.07621">Second best, Third worst, Fourth in line</a>, arxiv:2202.07621 [math.CO], 2022.
%F Sum_{k=0..n-2} k * T(n,k) = A332907(n) for n >= 3. - _Alois P. Heinz_, Dec 12 2021
%e Triangle begins:
%e [0] 1;
%e [1] 1;
%e [2] 2;
%e [3] 5, 1;
%e [4] 17, 1, 6;
%e [5] 74, 11, 15, 20;
%e [6] 394, 56, 60, 120, 90;
%e [7] 2484, 407, 525, 490, 630, 504;
%e [8] 18108, 3235, 4725, 2240, 4620, 4032, 3360;
%e [9] 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920;
%e ...
%p m:= infinity:
%p b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[3]=m,
%p 0, l[3]), add(b(n-j, sort([l[], j])[1..3])
%p *binomial(n-1, j-1)*(j-1)!, j=1..n))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$3])):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Dec 11 2021
%t m = Infinity;
%t b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[3]] == m, 0, l[[3]]], Sum[b[n-j, Sort[Append[l, j]][[1;;3]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
%t T[n_] := With[{p = b[n, {m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y Column 0 gives 1 together with A000774.
%Y Column 1 gives the column 3 of A208956.
%Y Row sums give A000142.
%Y Cf. A126074, A145877, A332907, A349979, A349980, A350015, A350273, A350274.
%K nonn,tabf
%O 0,3
%A _Steven Finch_, Dec 08 2021
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