A292508 Number A(n,k) of partitions of n with k kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 2, 1, 4, 7, 7, 5, 2, 1, 5, 11, 14, 12, 7, 4, 1, 6, 16, 25, 26, 19, 11, 4, 1, 7, 22, 41, 51, 45, 30, 15, 7, 1, 8, 29, 63, 92, 96, 75, 45, 22, 8, 1, 9, 37, 92, 155, 188, 171, 120, 67, 30, 12, 1, 10, 46, 129, 247, 343, 359, 291, 187, 97, 42, 14

Offset

0,8

Comments

Partial sum operator applied to column k gives column k+1.

A(n,k) is also defined for k < 0. All given formulas and programs can be applied also if k is negative.

Links

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

Formula

G.f. of column k: 1/(1-x)^k * 1/Product_{j>1} (1-x^j).

Column k is Euler transform of k,1,1,1,... .

For fixed k>=0, A(n,k) ~ 2^((k-5)/2) * 3^((k-2)/2) * n^((k-3)/2) * exp(Pi*sqrt(2*n/3)) / Pi^(k-1). - Vaclav Kotesovec, Oct 24 2018

Example

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,  2,  4,   7,  11,   16,   22,   29,    37, ...
  1,  3,  7,  14,  25,   41,   63,   92,   129, ...
  2,  5, 12,  26,  51,   92,  155,  247,   376, ...
  2,  7, 19,  45,  96,  188,  343,  590,   966, ...
  4, 11, 30,  75, 171,  359,  702, 1292,  2258, ...
  4, 15, 45, 120, 291,  650, 1352, 2644,  4902, ...
  7, 22, 67, 187, 478, 1128, 2480, 5124, 10026, ...

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+k-1)*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1,
      A(n, k+1)-A(n-1, k+1), `if`(k=1, combinat[numbpart](n),
      A(n-1, k)+A(n, k-1))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
# third Maple program:
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
    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 < 2, Binomial[k + n - 1, n], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 17 2018, translated from 3rd Maple program *)

Crossrefs

Columns k=0-10 give: A002865, A000041, A000070, A014153, A014160, A014161, A120477, A320753, A320754, A320755, A320756.

Rows n=0-4 give: A000012, A001477, A000124, A004006(k+1), A027927(k+3).

Main diagonal gives A292463.

A(n,n+1) gives A292613.

Cf. A292622, A292741, A292745.

Sequence in context: A115594 A086623 A248736 * A237597 A034928 A280267

Adjacent sequences: A292505 A292506 A292507 * A292509 A292510 A292511

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 17 2017

Status

approved