A185062 Number of n-element subsets that can be chosen from {1,2,...,2*n^3} having element sum n^4.

1, 1, 7, 351, 56217, 18878418, 11163952389, 10292468330630, 13703363417260677, 24932632800863823135, 59509756600504616529186, 180533923700628895521591343, 678854993880375551144618682344, 3100113915888360851262910882014885

Offset

0,3

Comments

a(n) is the number of partitions of n^4 into n distinct parts <= 2*n^3.

Links

Table of n, a(n) for n=0..13.

Example

a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.

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^4, 2*n^3, n):
seq(a(n), n=0..5);

Mathematica

$RecursionLimit = 10000;
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^4, 2 n^3, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 10}] (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

Crossrefs

Column k=3 of A185282.

Cf. A202261, A186730, A204459.

Sequence in context: A142669 A239717 A266876 * A067556 A082098 A057634

Adjacent sequences: A185059 A185060 A185061 * A185063 A185064 A185065

Keyword

nonn

Author

Alois P. Heinz, Jan 22 2012

Status

approved