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

%I A186730 #16 Dec 05 2020 09:42:12
%S A186730 1,1,3,36,785,26404,1235580,74394425,5503963083,484133307457,
%T A186730 49427802479445,5750543362215131,751453252349649771,
%U A186730 109016775078856564392,17391089152542558703435,3026419470005398093836960,570632810506646981058828349,115900277419940965862120360831
%N A186730 Number of n-element subsets that can be chosen from {1,2,...,2*n^2} having element sum n^3.
%C A186730 a(n) is the number of partitions of n^3 into n distinct parts <= 2*n^2.
%e A186730 a(0) = 1: {}.
%e A186730 a(1) = 1: {1}.
%e A186730 a(2) = 3: {1,7}, {2,6}, {3,5}.
%p A186730 b:= proc(n, i, t) option remember;
%p A186730       `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p A186730       `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p A186730     end:
%p A186730 a:= n-> b(n^3, 2*n^2, n):
%p A186730 seq(a(n), n=0..12);
%t A186730 $RecursionLimit = 2000;
%t A186730 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]]]];
%t A186730 a[n_] := b[n^3, 2n^2, n];
%t A186730 a /@ Range[0, 17] (* _Jean-François Alcover_, Dec 05 2020, after _Alois P. Heinz_ *)
%Y A186730 Column k=2 of A185282.
%Y A186730 Cf. A202261, A204459.
%K A186730 nonn
%O A186730 0,3
%A A186730 _Alois P. Heinz_, Jan 21 2012