A097592 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of even length.

1, 1, 1, 1, 2, 4, 7, 12, 5, 25, 52, 43, 102, 299, 258, 61, 531, 1750, 1853, 906, 3141, 11195, 15634, 8965, 1385, 20218, 83074, 133697, 94398, 31493, 146215, 675304, 1207256, 1088575, 460929, 50521, 1174889, 5880354, 11974457, 12625694, 6632158

Offset

0,5

Comments

Row n has 1+floor(n/2) entries.

Links

Alois P. Heinz, Rows n = 0..180, flattened

Formula

E.g.f.: 2(t-1)u/[ -2u+(2-t+tu)exp((-1+u)x/2)+(t-2+tu)exp(-(1+u)x/2)], where u=sqrt(5-4t).

Sum_{k=1..floor(n/2)} k * T(n,k) = A097593(n). - Alois P. Heinz, Jul 04 2019

Example

Triangle starts:
    1;
    1;
    1,   1;
    2,   4;
    7,  12,   5;
   25,  52,  43;
  102, 299, 258, 61;
Example: T(4,2) = 5 because we have 13/24, 14/23, 23/14, 24/13 and 34/12.

Maple

G:=2*(t-1)*u/(-2*u+(2-t+t*u)*exp((-1+u)*x/2)+(t-2+t*u)*exp(-(1+u)*x/2)): u:=sqrt(5-4*t): Gser:=simplify(series(G, x=0, 12)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(n!*coeff(Gser, x^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)), 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, 0)*x^t, j=1..u)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
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, 0]*x^t, {j, 1, u}]]]; T[n_] := Function[ {p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A097597, A317281, A317282, A317283, A317284, A317285, A317286, A317287, A317288, A317289, A317290.

Row sums give A000142.

T(n,floor(n/2)) gives A317139.

T(2n,n) gives A000364.

T(2n+1,n) gives A317140.

Cf. A097591, A097593.

Sequence in context: A058103 A379498 A346268 * A267699 A193841 A052474

Adjacent sequences: A097589 A097590 A097591 * A097593 A097594 A097595

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 29 2004

Status

approved