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A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

%I A350079 #34 Feb 17 2022 14:06:40
%S A350079 1,1,3,1,17,1,9,142,19,27,68,1569,201,135,510,710,21576,2921,3465,
%T A350079 2890,6390,9414,355081,50233,63630,20230,84490,98847,151032,6805296,
%U A350079 1004599,1196181,918680,705740,1493688,1812384,2840648,148869153,22872097,26904339,23943752,6351660,28072548,30810528,38348748,61247664
%N A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).
%C A350079 An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
%C A350079 If the mapping has no second component, then its second-smallest component is defined to have size 0.
%H A350079 Alois P. Heinz, <a href="/A350079/b350079.txt">Rows n = 0..141, flattened</a>
%H A350079 Steven Finch, <a href="http://arxiv.org/abs/2202.07621">Second best, Third worst, Fourth in line</a>, arxiv:2202.07621 [math.CO], 2022.
%e A350079 Triangle begins:
%e A350079        1;
%e A350079        1;
%e A350079        3,     1;
%e A350079       17,     1,     9;
%e A350079      142,    19,    27,    68;
%e A350079     1569,   201,   135,   510,   710;
%e A350079    21576,  2921,  3465,  2890,  6390,  9414;
%e A350079   355081, 50233, 63630, 20230, 84490, 98847, 151032;
%e A350079   ...
%p A350079 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350079 b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[2],
%p A350079       add(b(n-i, sort([l[], i])[1..2])*g(i)*binomial(n-1, i-1), i=1..n))
%p A350079     end:
%p A350079 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$2])):
%p A350079 seq(T(n), n=0..12);  # _Alois P. Heinz_, Dec 17 2021
%t A350079 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350079 b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[2]], Sum[b[n - i, Sort[Append[l, i]][[1;;2]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
%t A350079 T[n_] := With[{p = b[n, {Infinity, Infinity}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350079 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A350079 Column 0 gives gives 1 together with A001865.
%Y A350079 Row sums give A000312.
%Y A350079 Cf. A001865, A350078, A350080, A350081, A350275, A350276
%K A350079 nonn,tabf
%O A350079 0,3
%A A350079 _Steven Finch_, Dec 12 2021
%E A350079 More terms (two rows) from _Alois P. Heinz_, Dec 15 2021