A294219 Number T(n,k) of ascent sequences of length n where the maximum of 0 and all letter multiplicities equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 9, 4, 1, 0, 1, 26, 20, 5, 1, 0, 1, 82, 97, 30, 6, 1, 0, 1, 276, 496, 191, 42, 7, 1, 0, 1, 1014, 2686, 1259, 310, 56, 8, 1, 0, 1, 4006, 15481, 8784, 2416, 470, 72, 9, 1, 0, 1, 17046, 94843, 65012, 19787, 4141, 677, 90, 10, 1

Offset

0,9

Links

Table of n, a(n) for n=0..65.

Formula

T(n,k) = A294220(n,k) - A294220(n,k-1) for k>0, T(n,0) = A294220(n,k) = A000007(n).

Example

T(4,1) = 1: 0123.
T(4,2) = 9: 0011, 0012, 0101, 0102, 0110, 0112, 0120, 0121, 0122.
T(4,3) = 4: 0001, 0010, 0100, 0111.
T(4,4) = 1: 0000.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,     1;
  0, 1,     3,     1;
  0, 1,     9,     4,     1;
  0, 1,    26,    20,     5,     1;
  0, 1,    82,    97,    30,     6,    1;
  0, 1,   276,   496,   191,    42,    7,   1;
  0, 1,  1014,  2686,  1259,   310,   56,   8,  1;
  0, 1,  4006, 15481,  8784,  2416,  470,  72,  9,  1;
  0, 1, 17046, 94843, 65012, 19787, 4141, 677, 90, 10, 1;
  ...

Maple

b:= proc(n, i, t, p, k) option remember; `if`(n=0, 1,
      add(`if`(coeff(p, x, j)=k, 0, b(n-1, j, t+
          `if`(j>i, 1, 0), p+x^j, k)), j=1..t+1))
    end:
A:= (n, k)-> b(n, 0$3, k):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n == 0, 1, Sum[If[ Coefficient[p, x, j] == k, 0, b[n - 1, j, t + If[j > i, 1, 0], p + x^j, k]], {j, t + 1}]];
A[n_, k_] :=  b[n, 0, 0, 0, k];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *)

Crossrefs

Columns k=0-1 give: A000007, A057427.

Row sums give A022493.

Cf. A294220.

Sequence in context: A193357 A361953 A112413 * A091480 A034374 A261318

Adjacent sequences: A294216 A294217 A294218 * A294220 A294221 A294222

Keyword

nonn,tabl

Author

Alois P. Heinz, Oct 25 2017

Status

approved