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A350015 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/3).

%I A350015 #30 Feb 17 2022 13:20:40
%S A350015 1,1,2,5,1,17,7,74,46,394,311,15,2484,2241,315,18108,17627,4585,
%T A350015 149904,152839,57897,2240,1389456,1460944,705600,72800,14257440,
%U A350015 15326180,8673060,1660120,160460640,175421214,110271546,31600800,1247400,1965444480,2177730270,1469308698,559402272,55135080
%N A350015 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/3).
%C A350015 If the permutation has no third cycle, then its third-longest cycle is defined to have length 0.
%H A350015 Alois P. Heinz, <a href="/A350015/b350015.txt">Rows n = 0..140, flattened</a>
%H A350015 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 A350015 Sum_{k=0..floor(n/3)} k * T(n,k) = A332852(n) for n >= 3. - _Alois P. Heinz_, Dec 12 2021
%e A350015 Triangle begins:
%e A350015 [0]      1;
%e A350015 [1]      1;
%e A350015 [2]      2;
%e A350015 [3]      5,      1;
%e A350015 [4]     17,      7;
%e A350015 [5]     74,     46;
%e A350015 [6]    394,    311,    15;
%e A350015 [7]   2484,   2241,   315;
%e A350015 [8]  18108,  17627,  4585;
%e A350015 [9] 149904, 152839, 57897, 2240;
%e A350015 ...
%p A350015 b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
%p A350015       b(n-j, sort([l[], j])[2..4])*binomial(n-1, j-1), j=1..n))
%p A350015     end:
%p A350015 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
%p A350015 seq(lprint(T(n)), n=0..14);  # _Alois P. Heinz_, Dec 11 2021
%t A350015 b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[(j - 1)!*b[n - j, Sort[Append[l, j]][[2 ;; 4]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
%t A350015 T[n_] := With[{p = b[n, {0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350015 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A350015 Column 0 gives 1 together with A000774.
%Y A350015 Row sums give A000142.
%Y A350015 Cf. A126074, A145877, A332852, A349979, A349980, A350016, A350273, A350274.
%K A350015 nonn,tabf
%O A350015 0,3
%A A350015 _Steven Finch_, Dec 08 2021