A185062 - OEIS

%I #14 Mar 05 2021 06:13:39

%S 1,1,7,351,56217,18878418,11163952389,10292468330630,

%T 13703363417260677,24932632800863823135,59509756600504616529186,

%U 180533923700628895521591343,678854993880375551144618682344,3100113915888360851262910882014885

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

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

%e a(0) = 1: {}.

%e a(1) = 1: {1}.

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

%p b:= proc(n, i, t) option remember;

%p       `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,

%p       `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))

%p     end:

%p a:= n-> b(n^4, 2*n^3, n):

%p seq(a(n), n=0..5);

%t $RecursionLimit = 10000;

%t b[n_, i_, t_] := b[n, i, t] =

%t      If[i < t || n < t(t+1)/2 || n > t(2i - t + 1)/2, 0,

%t      If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]];

%t a[n_] := b[n^4, 2 n^3, n];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 10}] (* _Jean-François Alcover_, Mar 05 2021, after _Alois P. Heinz_ *)

%Y Column k=3 of A185282.

%Y Cf. A202261, A186730, A204459.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 22 2012