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A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A255212 #17 Mar 28 2023 21:08:47
%S A255212 1,1,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,1,1,1,0,1,1,2,2,1,1,0,1,1,2,2,1,2,
%T A255212 1,0,1,1,2,2,2,2,1,1,0,1,1,2,3,3,3,2,1,1,0,1,1,2,3,3,4,4,2,1,1,0,1,1,
%U A255212 2,3,4,5,5,4,1,1,1,0,1,1,2,4,5,7,9,6,2,4,2,1,0
%N A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A255212 Alois P. Heinz, <a href="/A255212/b255212.txt">Antidiagonals n = 0..140, flattened</a>
%e A255212 Square array A(n,k) begins:
%e A255212   1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
%e A255212   0, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
%e A255212   0, 1, 1, 1, 2,  2,  2,  2,  2,  2,  2, ...
%e A255212   0, 1, 1, 2, 2,  2,  3,  3,  3,  4,  4, ...
%e A255212   0, 1, 1, 1, 2,  3,  3,  4,  5,  5,  6, ...
%e A255212   0, 1, 2, 2, 3,  4,  5,  7,  8,  9, 11, ...
%e A255212   0, 1, 1, 2, 4,  5,  9, 10, 11, 15, 17, ...
%e A255212   0, 1, 1, 2, 4,  6,  9, 13, 18, 21, 27, ...
%e A255212   0, 1, 1, 1, 2,  7,  9, 16, 25, 30, 41, ...
%e A255212   0, 1, 1, 4, 6,  8, 18, 27, 36, 52, 68, ...
%e A255212   0, 1, 2, 2, 7, 13, 23, 36, 51, 70, 94, ...
%p A255212 b:= proc(n, i, t) option remember; `if`(n=0 or i=1 and n<=t, 1,
%p A255212       (j-> `if`(t*j<n, 0, b(n, i-1, t)+
%p A255212            `if`(j>n, 0, b(n-j, i, t-1))))(i^2))
%p A255212     end:
%p A255212 A:= (n, k)-> b(n^2, n, k):
%p A255212 seq(seq(A(n, d-n), n=0..d), d=0..15);
%t A255212 b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1 && n <= t, 1, Function[j, If[t*j<n, 0, b[n, i-1, t] + If[j>n, 0, b[n-j, i, t-1]]]][i^2]]; A[n_, k_] := b[n^2, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y A255212 Columns k=0-10 give: A000007, A000012, A063014, A016727, A065458, A065459, A065460, A065461, A065462, A255213, A255214.
%Y A255212 Main diagonal gives A105152.
%Y A255212 Cf. A302996.
%K A255212 nonn,tabl
%O A255212 0,24
%A A255212 _Alois P. Heinz_, Feb 17 2015