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A204461 Number of n-element subsets that can be chosen from {1,2,...,5*n} having element sum n*(5*n+1)/2.

%I A204461 #13 Dec 07 2020 02:07:42
%S A204461 1,1,5,25,177,1394,11963,108108,1016737,9853759,97809616,989878326,
%T A204461 10180316707,106124695130,1119148085092,11920389375524,
%U A204461 128077285062639,1386689101261013,15115933170815361,165776800325379769,1828006462946421194,20256667860779557632
%N A204461 Number of n-element subsets that can be chosen from {1,2,...,5*n} having element sum n*(5*n+1)/2.
%C A204461 a(n) is the number of partitions of n*(5*n+1)/2 into n distinct parts <=5*n.
%H A204461 Alois P. Heinz, <a href="/A204461/b204461.txt">Table of n, a(n) for n = 0..100</a>
%e A204461 a(2) = 5 because there are 5 2-element subsets that can be chosen from {1,2,...,10} having element sum 11: {1,10}, {2,9}, {3,8}, {4,7}, {5,6}.
%p A204461 b:= proc(n, i, t) option remember;
%p A204461       `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p A204461       `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p A204461     end:
%p A204461 a:= n-> b(n*(5*n+1)/2, 5*n, n):
%p A204461 seq(a(n), n=0..20);
%t A204461 b[n_, i_, t_] /; i<t || n<t(t+1)/2 || n>t(2i-t+1)/2 = 0; b[0, _, _] = 1;
%t A204461 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 A204461 a[n_] := b[n(5n+1)/2, 5n, n];
%t A204461 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A204461 Row n=5 of A204459.
%K A204461 nonn
%O A204461 0,3
%A A204461 _Alois P. Heinz_, Jan 18 2012