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A340626 a(n) = Sum_{d|n, d odd} binomial(d+n/d-1, d).

%I A340626 #25 Apr 26 2021 08:01:15
%S A340626 1,2,4,4,6,10,8,8,20,16,12,32,14,22,72,16,18,84,20,76,142,34,24,144,
%T A340626 152,40,248,148,30,518,32,32,398,52,828,620,38,58,600,832,42,1416,44,
%U A340626 408,2864,70,48,864,1766,2078,1192,612,54,3224,4424,3488,1598,88,60,6784,62,94,13528,64,8634
%N A340626 a(n) = Sum_{d|n, d odd} binomial(d+n/d-1, d).
%H A340626 Seiichi Manyama, <a href="/A340626/b340626.txt">Table of n, a(n) for n = 1..10000</a>
%F A340626 G.f.: (1/2) * Sum_{k >= 1} (1/(1 - x^k)^k - 1/(1 + x^k)^k).
%F A340626 G.f.: Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1))^(2*k).
%F A340626 a(n) = (A081543(n) + A338682(n))/2.
%F A340626 If p is prime, a(p) = (p mod 2) + p.
%t A340626 a[n_] := DivisorSum[n, Binomial[# + n/# - 1, #] &, OddQ[#] &]; Array[a, 65] (* _Amiram Eldar_, Apr 25 2021 *)
%o A340626 (PARI) a(n) = sumdiv(n, d, (d%2)*binomial(d+n/d-1, d));
%o A340626 (PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-x^k)^k-1/(1+x^k)^k)/2)
%o A340626 (PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(2*k-1))^(2*k)))
%Y A340626 Cf. A081543, A338682, A340625.
%K A340626 nonn
%O A340626 1,2
%A A340626 _Seiichi Manyama_, Apr 25 2021