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A204465 Number of n-element subsets that can be chosen from {1,2,...,9*n} having element sum n*(9*n+1)/2.

%I A204465 #11 Dec 07 2020 02:08:12
%S A204465 1,1,9,85,1143,17053,276373,4721127,83916031,1537408202,28851490163,
%T A204465 552095787772,10736758952835,211657839534446,4221164530621965,
%U A204465 85031286025167082,1727896040082882283,35382865902724442331,729502230296220422918,15132164184348997874504
%N A204465 Number of n-element subsets that can be chosen from {1,2,...,9*n} having element sum n*(9*n+1)/2.
%C A204465 a(n) is the number of partitions of n*(9*n+1)/2 into n distinct parts <=9*n.
%e A204465 a(2) = 9 because there are 9 2-element subsets that can be chosen from {1,2,...,18} having element sum 19: {1,18}, {2,17}, {3,16}, {4,15}, {5,14}, {6,13}, {7,12}, {8,11}, {9,10}.
%p A204465 b:= proc(n, i, t) option remember;
%p A204465       `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p A204465       `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p A204465     end:
%p A204465 a:= n-> b(n*(9*n+1)/2, 9*n, n):
%p A204465 seq(a(n), n=0..20);
%t A204465 b[n_, i_, t_] /; i<t || n<t(t+1)/2 || n>t(2i-t+1)/2 = 0; b[0, _, _] = 1;
%t A204465 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 A204465 a[n_] := b[n(9n+1)/2, 9n, n];
%t A204465 a /@ Range[0, 10] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A204465 Row n=9 of A204459.
%K A204465 nonn
%O A204465 0,3
%A A204465 _Alois P. Heinz_, Jan 18 2012