A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 6, 8, 5, 0, 1, 5, 8, 15, 14, 7, 0, 1, 6, 10, 24, 27, 24, 11, 0, 1, 7, 12, 35, 44, 51, 40, 15, 0, 1, 8, 14, 48, 65, 88, 93, 64, 22, 0, 1, 9, 16, 63, 90, 135, 176, 159, 100, 30, 0, 1, 10, 18, 80, 119, 192, 295, 312, 264, 154, 42, 0

Offset

0,8

Links

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Jessica Jay and Benjamin Lees, Combinatorial identities from an inhomogeneous Ising chain, arXiv:2401.16311 [math.PR], 2024.

Wikipedia, Partition (number theory)

Formula

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

A(n,k) = Sum_{i=0..floor((sqrt(1+8*k)-1)/2)} k!/(k-i)! * A321878(n,i).

Example

A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b.
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,    1,    1,    1, ...
  0,  1,   2,   3,   4,   5,    6,    7,    8, ...
  0,  2,   4,   6,   8,  10,   12,   14,   16, ...
  0,  3,   8,  15,  24,  35,   48,   63,   80, ...
  0,  5,  14,  27,  44,  65,   90,  119,  152, ...
  0,  7,  24,  51,  88, 135,  192,  259,  336, ...
  0, 11,  40,  93, 176, 295,  456,  665,  928, ...
  0, 15,  64, 159, 312, 535,  840, 1239, 1744, ...
  0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/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, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/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, Apr 30 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-4 give: A000007, A000041, A015128, A264686, A266821.

Main diagonal gives A321880.

Cf, A000142, A003056, A321878.

Sequence in context: A059259 A124394 A086460 * A136431 A182888 A317205

Adjacent sequences: A321881 A321882 A321883 * A321885 A321886 A321887

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 27 2019

Status

approved