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A185282 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of n-element subsets that can be chosen from {1,2,...,2*n^k} having element sum n^(k+1).

%I A185282 #18 Sep 21 2018 22:45:17
%S A185282 1,1,1,1,1,0,1,1,1,0,1,1,3,3,0,1,1,7,36,7,0,1,1,15,351,785,18,0,1,1,
%T A185282 31,3240,56217,26404,51,0,1,1,63,29403,3695545,18878418,1235580,155,0,
%U A185282 1,1,127,265356,238085177,12107973904,11163952389,74394425,486,0
%N A185282 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of n-element subsets that can be chosen from {1,2,...,2*n^k} having element sum n^(k+1).
%C A185282 A(n,k) is the number of partitions of n^(k+1) into n distinct parts <= 2*n^k.
%e A185282 A(0,0) = 1: {}.
%e A185282 A(1,1) = 1: {1}.
%e A185282 A(2,2) = 3: {1,7}, {2,6}, {3,5}.
%e A185282 A(3,1) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
%e A185282 A(4,1) = 7: {1,2,5,8}, {1,2,6,7}, {1,3,4,8}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}.
%e A185282 A(2,3) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.
%e A185282 Square array A(n,k) begins:
%e A185282   1,   1,      1,         1,            1, ...
%e A185282   1,   1,      1,         1,            1, ...
%e A185282   0,   1,      3,         7,           15, ...
%e A185282   0,   3,     36,       351,         3240, ...
%e A185282   0,   7,    785,     56217,      3695545, ...
%e A185282   0,  18,  26404,  18878418,  12107973904, ...
%p A185282 b:= proc(n, i, t) option remember;
%p A185282       `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p A185282       `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p A185282     end:
%p A185282 A:= (n, k)-> b(n^(k+1), 2*n^k, n):
%p A185282 seq(seq(A(n, d-n), n=0..d), d=0..8);
%t A185282 $RecursionLimit = 10000; b[n_, i_, t_] := b[n, i, t] = If [i < t || n < t*(t+1)/2 || n > t*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]]; A[0, _] = A[1, _] = 1; A[n_ /; n > 1, 0] = 0; A[n_, k_] := b[n^(k+1), 2*n^k, n]; Table[Print[ta = Table [A[n, d-n], {n, 0, d}]]; ta, {d, 0, 9}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A185282 Rows n=1-3 give: A000012, A000225, A026121.
%Y A185282 Columns k=1-3 give: A202261, A186730, A185062.
%K A185282 nonn,tabl
%O A185282 0,13
%A A185282 _Alois P. Heinz_, Jan 25 2012