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%I #17 Mar 28 2023 21:08:47
%S 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 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 2,3,4,5,5,4,1,1,1,0,1,1,2,4,5,7,9,6,2,4,2,1,0
%N 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 Alois P. Heinz, <a href="/A255212/b255212.txt">Antidiagonals n = 0..140, flattened</a>
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, ...
%e 0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, ...
%e 0, 1, 1, 1, 2, 3, 3, 4, 5, 5, 6, ...
%e 0, 1, 2, 2, 3, 4, 5, 7, 8, 9, 11, ...
%e 0, 1, 1, 2, 4, 5, 9, 10, 11, 15, 17, ...
%e 0, 1, 1, 2, 4, 6, 9, 13, 18, 21, 27, ...
%e 0, 1, 1, 1, 2, 7, 9, 16, 25, 30, 41, ...
%e 0, 1, 1, 4, 6, 8, 18, 27, 36, 52, 68, ...
%e 0, 1, 2, 2, 7, 13, 23, 36, 51, 70, 94, ...
%p b:= proc(n, i, t) option remember; `if`(n=0 or i=1 and n<=t, 1,
%p (j-> `if`(t*j<n, 0, b(n, i-1, t)+
%p `if`(j>n, 0, b(n-j, i, t-1))))(i^2))
%p end:
%p A:= (n, k)-> b(n^2, n, k):
%p seq(seq(A(n, d-n), n=0..d), d=0..15);
%t 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 Columns k=0-10 give: A000007, A000012, A063014, A016727, A065458, A065459, A065460, A065461, A065462, A255213, A255214.
%Y Main diagonal gives A105152.
%Y Cf. A302996.
%K nonn,tabl
%O 0,24
%A _Alois P. Heinz_, Feb 17 2015
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