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

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

Author

Seiichi Manyama, Apr 25 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A081543, A338682, A340625.

Programs

Mathematica

a[n_] := DivisorSum[n, Binomial[# + n/# - 1, #] &, OddQ[#] &]; Array[a, 65] (* Amiram Eldar, Apr 25 2021 *)

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)))

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.