A292741 Number A(n,k) of partitions of n with k sorts of part 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, 5, 3, 2, 1, 4, 10, 11, 5, 2, 1, 5, 17, 31, 24, 7, 4, 1, 6, 26, 69, 95, 50, 11, 4, 1, 7, 37, 131, 278, 287, 104, 15, 7, 1, 8, 50, 223, 657, 1114, 865, 212, 22, 8, 1, 9, 65, 351, 1340, 3287, 4460, 2599, 431, 30, 12, 1, 10, 82, 521, 2459, 8042, 16439, 17844, 7804, 870, 42, 14

Offset

0,8

Links

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

Formula

G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).

A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).

Example

A(1,3) = 3: 1a, 1b, 1c.
A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c.
A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,      1,      1, ...
  0,  1,   2,    3,     4,     5,      6,      7, ...
  1,  2,   5,   10,    17,    26,     37,     50, ...
  1,  3,  11,   31,    69,   131,    223,    351, ...
  2,  5,  24,   95,   278,   657,   1340,   2459, ...
  2,  7,  50,  287,  1114,  3287,   8042,  17215, ...
  4, 11, 104,  865,  4460, 16439,  48256, 120509, ...
  4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n,
      add(b(n-i*j, i-1, k), j=0..iquo(n, i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[0, _, _] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[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 19 2018, translated from Maple *)

Crossrefs

Columns k=0-2 give: A002865, A000041, A090764.

Rows n=0-2 give: A000012, A001477, A002522, A071568.

Main diagonal gives A292462.

Cf. A003992, A004248, A009998, A051129, A292508, A292622, A292745.

Sequence in context: A034254 A157103 A135966 * A356802 A060351 A335845

Adjacent sequences: A292738 A292739 A292740 * A292742 A292743 A292744

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 22 2017

Status

approved