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A258797 a(n) = [x^n] Product_{k=1..n} (1+x^k)^2 / x^k.

%I A258797 #18 Jul 14 2025 17:08:06
%S A258797 1,1,2,6,16,51,166,554,1896,6595,23212,82582,296393,1071738,3900696,
%T A258797 14278074,52526972,194108087,720197524,2681854490,10019539112,
%U A258797 37545876368,141080872362,531457445806,2006678785762,7593123695669,28789152013570,109356019134584
%N A258797 a(n) = [x^n] Product_{k=1..n} (1+x^k)^2 / x^k.
%C A258797 a(n) is half the number of subsets of {-n..n} whose sum is n. - _Ilya Gutkovskiy_, Jul 09 2025
%H A258797 Alois P. Heinz, <a href="/A258797/b258797.txt">Table of n, a(n) for n = 0..555</a>
%F A258797 a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)).
%p A258797 b:= proc(n, s) option remember; `if`(n*(n+1)/2<s, 0, `if`(n=0, 1,
%p A258797       add(`if`(j=0, 2, 1)*b(n-1, abs(s+j*n)), j=-1..1)))
%p A258797     end:
%p A258797 a:= n-> b(n$2):
%p A258797 seq(a(n), n=0..27);  # _Alois P. Heinz_, Jul 14 2025
%t A258797 Table[SeriesCoefficient[Product[(1+x^k)^2/x^k, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
%t A258797 Table[SeriesCoefficient[Product[1+x^k, {k, 1, n}]^2, {x, 0, n*(n+3)/2}], {n, 0, 30}]
%Y A258797 Cf. A022567, A047653, A258798, A258799.
%K A258797 nonn
%O A258797 0,3
%A A258797 _Vaclav Kotesovec_, Jun 10 2015