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A097591 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of odd length.

%I A097591 #28 Feb 22 2024 20:26:44
%S A097591 1,0,1,1,0,1,0,5,0,1,6,0,17,0,1,0,70,0,49,0,1,90,0,500,0,129,0,1,0,
%T A097591 1890,0,2828,0,321,0,1,2520,0,23100,0,13930,0,769,0,1,0,83160,0,
%U A097591 215292,0,62634,0,1793,0,1,113400,0,1549800,0,1697430,0,264072,0,4097,0,1
%N A097591 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of odd length.
%H A097591 Alois P. Heinz, <a href="/A097591/b097591.txt">Rows n = 0..140, flattened</a>
%F A097591 E.g.f.: t^2/[1-tx-(1-t^2)exp(-tx)].
%F A097591 Sum_{k=1..n} k * T(n,k) = A096654(n-1) for n > 0. - _Alois P. Heinz_, Jul 03 2019
%e A097591 Triangle starts:
%e A097591      1;
%e A097591      0,    1;
%e A097591      1,    0,     1;
%e A097591      0,    5,     0,    1;
%e A097591      6,    0,    17,    0,     1;
%e A097591      0,   70,     0,   49,     0,   1;
%e A097591     90,    0,   500,    0,   129,   0,   1;
%e A097591      0, 1890,     0, 2828,     0, 321,   0, 1;
%e A097591   2520,    0, 23100,    0, 13930,   0, 769, 0, 1;
%e A097591   ...
%e A097591 Row n has n+1 entries.
%e A097591 Example: T(3,1) = 5 because we have (123), 13(2), (2)13, 23(1) and (3)12 (the runs of odd length are shown between parentheses).
%p A097591 G:=t^2/(1-t*x-(1-t^2)*exp(-t*x)): Gser:=simplify(series(G,x=0,12)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(expand(n!*coeff(Gser,x^n))) od: seq(seq(coeff(t*P[n],t^k),k=1..n+1),n=0..11);
%p A097591 # second Maple program:
%p A097591 b:= proc(u, o, t) option remember; `if`(u+o=0, x^t, expand(
%p A097591       add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+
%p A097591       add(b(u-j, o+j-1, 1)*x^t, j=1..u)))
%p A097591     end:
%p A097591 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0, 1)):
%p A097591 seq(T(n), n=0..12);  # _Alois P. Heinz_, Nov 19 2013
%t A097591 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, x^t, Expand[Sum[b[u+j-1, o-j, Mod[t+1, 2]], {j, 1, o}] + Sum[b[u-j, o+j-1, 1]*x^t, {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 19 2015, after _Alois P. Heinz_ *)
%Y A097591 Bisections of columns k=0-1 give: A000680, A302910.
%Y A097591 Row sums give A000142.
%Y A097591 T(n+1,n-1) gives A000337.
%Y A097591 T(4n,2n) gives A308962.
%Y A097591 Cf. A096654, A097592.
%K A097591 nonn,tabl
%O A097591 0,8
%A A097591 _Emeric Deutsch_, Aug 29 2004