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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

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-Fran├žois 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

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Last modified August 10 19:52 EDT 2020. Contains 336381 sequences. (Running on oeis4.)