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

%I A185062 #14 Mar 05 2021 06:13:39
%S A185062 1,1,7,351,56217,18878418,11163952389,10292468330630,
%T A185062 13703363417260677,24932632800863823135,59509756600504616529186,
%U A185062 180533923700628895521591343,678854993880375551144618682344,3100113915888360851262910882014885
%N A185062 Number of n-element subsets that can be chosen from {1,2,...,2*n^3} having element sum n^4.
%C A185062 a(n) is the number of partitions of n^4 into n distinct parts <= 2*n^3.
%e A185062 a(0) = 1: {}.
%e A185062 a(1) = 1: {1}.
%e A185062 a(2) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.
%p A185062 b:= proc(n, i, t) option remember;
%p A185062       `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p A185062       `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p A185062     end:
%p A185062 a:= n-> b(n^4, 2*n^3, n):
%p A185062 seq(a(n), n=0..5);
%t A185062 $RecursionLimit = 10000;
%t A185062 b[n_, i_, t_] := b[n, i, t] =
%t A185062      If[i < t || n < t(t+1)/2 || n > t(2i - t + 1)/2, 0,
%t A185062      If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]];
%t A185062 a[n_] := b[n^4, 2 n^3, n];
%t A185062 Table[Print[n, " ", a[n]]; a[n], {n, 0, 10}] (* _Jean-François Alcover_, Mar 05 2021, after _Alois P. Heinz_ *)
%Y A185062 Column k=3 of A185282.
%Y A185062 Cf. A202261, A186730, A204459.
%K A185062 nonn
%O A185062 0,3
%A A185062 _Alois P. Heinz_, Jan 22 2012