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

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, 1890, 0, 2828, 0, 321, 0, 1, 2520, 0, 23100, 0, 13930, 0, 769, 0, 1, 0, 83160, 0, 215292, 0, 62634, 0, 1793, 0, 1, 113400, 0, 1549800, 0, 1697430, 0, 264072, 0, 4097, 0, 1

Offset

0,8

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

E.g.f.: t^2/[1-tx-(1-t^2)exp(-tx)].

Sum_{k=1..n} k * T(n,k) = A096654(n-1) for n > 0. - Alois P. Heinz, Jul 03 2019

Example

Triangle starts:
     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, 1890,     0, 2828,     0, 321,   0, 1;
  2520,    0, 23100,    0, 13930,   0, 769, 0, 1;
  ...
Row n has n+1 entries.
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).

Maple

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);
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, x^t, expand(
      add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+
      add(b(u-j, o+j-1, 1)*x^t, j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 19 2013

Mathematica

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 *)

Crossrefs

Bisections of columns k=0-1 give: A000680, A302910.

Row sums give A000142.

T(n+1,n-1) gives A000337.

T(4n,2n) gives A308962.

Cf. A096654, A097592.

Sequence in context: A326185 A293508 A083861 * A318299 A164652 A127557

Adjacent sequences: A097588 A097589 A097590 * A097592 A097593 A097594

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 29 2004

Status

approved