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A340626
a(n) = Sum_{d|n, d odd} binomial(d+n/d-1, d).
2
1, 2, 4, 4, 6, 10, 8, 8, 20, 16, 12, 32, 14, 22, 72, 16, 18, 84, 20, 76, 142, 34, 24, 144, 152, 40, 248, 148, 30, 518, 32, 32, 398, 52, 828, 620, 38, 58, 600, 832, 42, 1416, 44, 408, 2864, 70, 48, 864, 1766, 2078, 1192, 612, 54, 3224, 4424, 3488, 1598, 88, 60, 6784, 62, 94, 13528, 64, 8634
OFFSET
1,2
LINKS
FORMULA
G.f.: (1/2) * Sum_{k >= 1} (1/(1 - x^k)^k - 1/(1 + x^k)^k).
G.f.: Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1))^(2*k).
a(n) = (A081543(n) + A338682(n))/2.
If p is prime, a(p) = (p mod 2) + p.
MATHEMATICA
a[n_] := DivisorSum[n, Binomial[# + n/# - 1, #] &, OddQ[#] &]; Array[a, 65] (* Amiram Eldar, Apr 25 2021 *)
PROG
(PARI) a(n) = sumdiv(n, d, (d%2)*binomial(d+n/d-1, d));
(PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-x^k)^k-1/(1+x^k)^k)/2)
(PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(2*k-1))^(2*k)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 25 2021
STATUS
approved