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A242820
Number T(n,k) of permutations of [n] with exactly k occurrences of the consecutive step pattern up, down, down, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows.
4
1, 1, 2, 6, 24, 116, 4, 672, 48, 4536, 504, 34944, 5376, 302896, 59488, 496, 2916992, 697856, 13952, 30899616, 8720448, 296736, 357080064, 116109312, 5812224, 4470310976, 1645662912, 110697408, 349504, 60269056512, 24776769024, 2114735616, 17730048
OFFSET
0,3
LINKS
EXAMPLE
T(5,1) = 4: (1,5,4,3,2), (2,5,4,3,1), (3,5,4,2,1), (4,5,3,2,1).
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 6;
: 4 : 24;
: 5 : 116, 4;
: 6 : 672, 48;
: 7 : 4536, 504;
: 8 : 34944, 5376;
: 9 : 302896, 59488, 496;
: 10 : 2916992, 697856, 13952;
: 11 : 30899616, 8720448, 296736;
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
add(b(u-j, o+j-1, [1, 3, 4, 1][t])*`if`(t=4, x, 1), j=1..u)+
add(b(u+j-1, o-j, 2), j=1..o)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Expand[
Sum[b[u - j, o + j - 1, {1, 3, 4, 1}[[t]]]*If[t==4, x, 1], {j, 1, u}]+
Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
T[n_] := CoefficientList[b[n, 0, 1], x];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, Mar 23 2021, after Alois P. Heinz *)
CROSSREFS
Column k=0 gives A177518.
Row sums give: A000142.
Sequence in context: A069657 A369766 A211321 * A228395 A082631 A212198
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 23 2014
STATUS
approved