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A204463 Number of n-element subsets that can be chosen from {1,2,...,7*n} having element sum n*(7*n+1)/2.

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