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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
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.
Sequence in context: A326185 A293508 A083861 * A318299 A164652 A127557
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Aug 29 2004
STATUS
approved

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Last modified June 24 11:41 EDT 2024. Contains 373677 sequences. (Running on oeis4.)