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%I #13 Dec 07 2020 02:07:54
%S 1,1,7,50,519,5910,73294,957332,13011585,182262067,2615047418,
%T 38257201350,568784501596,8571868074560,130687117401934,
%U 2012485947249822,31262279693472267,489374243181858825,7712880007117038531,122301036027089010734,1949904188227477978314
%N Number of n-element subsets that can be chosen from {1,2,...,7*n} having element sum n*(7*n+1)/2.
%C a(n) is the number of partitions of n*(7*n+1)/2 into n distinct parts <=7*n.
%H Alois P. Heinz, <a href="/A204463/b204463.txt">Table of n, a(n) for n = 0..80</a>
%e a(2) = 7 because there are 7 2-element subsets that can be chosen from {1,2,...,14} having element sum 15: {1,14}, {2,13}, {3,12}, {4,11}, {5,10}, {6,9}, {7,8}.
%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*(7*n+1)/2, 7*n, n):
%p seq(a(n), n=0..20);
%t b[n_, i_, t_] /; i<t || n<t(t+1)/2 || n>t(2i-t+1)/2 = 0; b[0, _, _] = 1;
%t b[n_, i_, t_] := b[n, i, t] = b[n, i-1, t] + If[n<i, 0, b[n-i, i-1, t-1]];
%t a[n_] := b[n(7n+1)/2, 7n, n];
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y Row n=7 of A204459.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jan 18 2012
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