A186730 Number of n-element subsets that can be chosen from {1,2,...,2*n^2} having element sum n^3.

1, 1, 3, 36, 785, 26404, 1235580, 74394425, 5503963083, 484133307457, 49427802479445, 5750543362215131, 751453252349649771, 109016775078856564392, 17391089152542558703435, 3026419470005398093836960, 570632810506646981058828349, 115900277419940965862120360831

Offset

0,3

Comments

a(n) is the number of partitions of n^3 into n distinct parts <= 2*n^2.

Links

Table of n, a(n) for n=0..17.

Example

a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 3: {1,7}, {2,6}, {3,5}.

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-> b(n^3, 2*n^2, n):
seq(a(n), n=0..12);

Mathematica

$RecursionLimit = 2000;
b[n_, i_, t_] := b[n, i, t] = If[i<t || n<t (t+1)/2 || n>t (2i-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[n_] := b[n^3, 2n^2, n];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

Crossrefs

Column k=2 of A185282.

Cf. A202261, A204459.

Sequence in context: A144758 A301582 A122220 * A224347 A227251 A326273

Adjacent sequences: A186727 A186728 A186729 * A186731 A186732 A186733

Keyword

nonn

Author

Alois P. Heinz, Jan 21 2012

Status

approved