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

0,8

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

Links

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

Formula

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

Example

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, ...

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 *)

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.

Sequence in context: A304382 A304717 A110963 * A292869 A106348 A161092

Adjacent sequences: A292619 A292620 A292621 * A292623 A292624 A292625

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 20 2017

Status

approved