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%I #11 Dec 07 2020 02:07:48
%S 1,6,318,32134,4083008,587267282,91403537276,15027205920330,
%T 2572042542065646,454018964549333284,82122490665668040962,
%U 15150820045467016057500,2841258381788564812646472,540201085284535788002286246,103917818379993516623446237348
%N Number of 2*n-element subsets that can be chosen from {1,2,...,12*n} having element sum n*(12*n+1).
%C a(n) is the number of partitions of n*(12*n+1) into 2*n distinct parts <=12*n.
%e a(1) = 6 because there are 6 2-element subsets that can be chosen from {1,2,...,12} having element sum 13: {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}.
%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*(12*n+1), 12*n, 2*n):
%p seq(a(n), n=0..12);
%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(12n+1), 12n, 2n];
%t a /@ Range[0, 10] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y Bisection of row n=6 of A204459.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Jan 18 2012
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