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A340625 a(n) = Sum_{d|n, d odd, d <= n/d} binomial(n/d, d).

%I A340625 #25 Apr 26 2021 08:01:11
%S A340625 1,2,3,4,5,6,7,8,10,10,11,16,13,14,25,16,17,38,19,20,56,22,23,80,26,
%T A340625 26,111,28,29,156,31,32,198,34,56,256,37,38,325,96,41,406,43,44,626,
%U A340625 46,47,608,50,302,731,52,53,870,517,64,1026,58,59,1992,61,62,1429,64,1352,1606,67,68,1840,2192,71,2096,73,74
%N A340625 a(n) = Sum_{d|n, d odd, d <= n/d} binomial(n/d, d).
%H A340625 Seiichi Manyama, <a href="/A340625/b340625.txt">Table of n, a(n) for n = 1..10000</a>
%F A340625 G.f.: (1/2) * Sum_{k >= 1} ((1 + x^k)^k - (1 - x^k)^k).
%F A340625 If p is prime, a(p) = p.
%t A340625 a[n_] := DivisorSum[n, Binomial[n/#, #] &, OddQ[#] &]; Array[a, 75] (* _Amiram Eldar_, Apr 25 2021 *)
%o A340625 (PARI) a(n) = sumdiv(n, d, (d%2)*binomial(n/d, d));
%o A340625 (PARI) N=99; x='x+O('x^N); Vec(sum(k=1, N, (1+x^k)^k-(1-x^k)^k)/2)
%Y A340625 Cf. A318636, A327124, A340626.
%K A340625 nonn
%O A340625 1,2
%A A340625 _Seiichi Manyama_, Apr 25 2021