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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.

%I A321884 #57 Jan 30 2024 04:39:11
%S A321884 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,
%T A321884 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,
%U A321884 63,90,135,176,159,100,30,0,1,10,18,80,119,192,295,312,264,154,42,0
%N 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.
%H A321884 Alois P. Heinz, <a href="/A321884/b321884.txt">Antidiagonals n = 0..200, flattened</a>
%H A321884 Jessica Jay and Benjamin Lees, <a href="https://arxiv.org/abs/2401.16311">Combinatorial identities from an inhomogeneous Ising chain</a>, arXiv:2401.16311 [math.PR], 2024.
%H A321884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A321884 G.f. of column k: Product_{j>=1} (1+(k-1)*x^j)/(1-x^j).
%F A321884 A(n,k) = Sum_{i=0..floor((sqrt(1+8*k)-1)/2)} k!/(k-i)! * A321878(n,i).
%e A321884 A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b.
%e A321884 Square array A(n,k) begins:
%e A321884   1,  1,   1,   1,   1,   1,    1,    1,    1, ...
%e A321884   0,  1,   2,   3,   4,   5,    6,    7,    8, ...
%e A321884   0,  2,   4,   6,   8,  10,   12,   14,   16, ...
%e A321884   0,  3,   8,  15,  24,  35,   48,   63,   80, ...
%e A321884   0,  5,  14,  27,  44,  65,   90,  119,  152, ...
%e A321884   0,  7,  24,  51,  88, 135,  192,  259,  336, ...
%e A321884   0, 11,  40,  93, 176, 295,  456,  665,  928, ...
%e A321884   0, 15,  64, 159, 312, 535,  840, 1239, 1744, ...
%e A321884   0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...
%p A321884 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A321884       (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
%p A321884     end:
%p A321884 A:= (n, k)-> b(n$2, k):
%p A321884 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A321884 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]]];
%t A321884 A[n_, k_] := b[n, n, k];
%t A321884 Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y A321884 Columns k=0-4 give: A000007, A000041, A015128, A264686, A266821.
%Y A321884 Main diagonal gives A321880.
%Y A321884 Cf, A000142, A003056, A321878.
%K A321884 nonn,tabl
%O A321884 0,8
%A A321884 _Alois P. Heinz_, Aug 27 2019