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).

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, 31, 3240, 56217, 26404, 51, 0, 1, 1, 63, 29403, 3695545, 18878418, 1235580, 155, 0, 1, 1, 127, 265356, 238085177, 12107973904, 11163952389, 74394425, 486, 0

Offset

0,13

Comments

A(n,k) is the number of partitions of n^(k+1) into n distinct parts <= 2*n^k.

Links

Table of n, a(n) for n=0..54.

Example

A(0,0) = 1: {}.
A(1,1) = 1: {1}.
A(2,2) = 3: {1,7}, {2,6}, {3,5}.
A(3,1) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
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}.
A(2,3) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.
Square array A(n,k) begins:
  1,   1,      1,         1,            1, ...
  1,   1,      1,         1,            1, ...
  0,   1,      3,         7,           15, ...
  0,   3,     36,       351,         3240, ...
  0,   7,    785,     56217,      3695545, ...
  0,  18,  26404,  18878418,  12107973904, ...

Maple

b:= proc(n, i, t) option remember;
      `if`(i<t or n<t*(t+1)/2 or 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))))
    end:
A:= (n, k)-> b(n^(k+1), 2*n^k, n):
seq(seq(A(n, d-n), n=0..d), d=0..8);

Mathematica

$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 *)

Crossrefs

Rows n=1-3 give: A000012, A000225, A026121.

Columns k=1-3 give: A202261, A186730, A185062.

Sequence in context: A288644 A172169 A306629 * A193470 A102752 A104548

Adjacent sequences: A185279 A185280 A185281 * A185283 A185284 A185285

Keyword

nonn,tabl

Author

Alois P. Heinz, Jan 25 2012

Status

approved