A292622 Number A(n,k) of partitions of n with up to k distinct kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 1, 4, 4, 3, 3, 2, 1, 5, 7, 5, 5, 4, 4, 1, 6, 11, 9, 8, 7, 6, 4, 1, 7, 16, 16, 13, 12, 10, 8, 7, 1, 8, 22, 27, 22, 20, 17, 14, 11, 8, 1, 9, 29, 43, 38, 33, 29, 24, 19, 15, 12, 1, 10, 37, 65, 65, 55, 49, 41, 33, 26, 20, 14

Offset: 0

Author

Alois P. Heinz, Sep 20 2017

Keywords

• nonn
• tabl

Comments

For fixed k>=0, A(n,k) ~ Pi * 2^(k - 5/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - Vaclav Kotesovec, Oct 24 2018

Examples

			A(3,4) =  9: 3, 21a, 21b, 21c, 21d, 1a1b1c, 1a1b1d, 1a1c1d, 1b1c1d.
A(4,3) =  8: 4, 31a, 31b, 31c, 22, 21a1b, 21a1c, 21b1c.
A(4,4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,  5,   6,   7,   8, ...
  1,  1,  2,  4,  7, 11,  16,  22,  29, ...
  1,  2,  3,  5,  9, 16,  27,  43,  65, ...
  2,  3,  5,  8, 13, 22,  38,  65, 108, ...
  2,  4,  7, 12, 20, 33,  55,  93, 158, ...
  4,  6, 10, 17, 29, 49,  82, 137, 230, ...
  4,  8, 14, 24, 41, 70, 119, 201, 338, ...
  7, 11, 19, 33, 57, 98, 168, 287, 488, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A002865, A027336, A320689, A320690, A320691, A320692, A320693, A320694, A320695, A320696, A320697.

Rows n=0-4 give: A000012, A001477, A000124(k-1) for k>0, A011826 for k>0.

Main diagonal gives A292507.

Cf. A292508, A292741, A292745.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1,
      binomial(k, n), `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

Formula

G.f. of column k: (1 + x)^k * Product_{j>=2} 1 / (1 - x^j). - Ilya Gutkovskiy, Apr 24 2021