A382503 a(n) = Sum_{d|n} binomial(2*d-1,d).

1, 4, 11, 39, 127, 476, 1717, 6474, 24321, 92508, 352717, 1352589, 5200301, 20060020, 77558897, 300546669, 1166803111, 4537592436, 17672631901, 68923356953, 269128938947, 1052049834580, 4116715363801, 16123803200574, 63205303219003, 247959271674356

Offset: 1

Author

Ilya Gutkovskiy, Apr 10 2025

Keywords

• nonn

Crossrefs

Cf. A000005, A000984, A001700, A045630, A072929, A088218.

Programs

Mathematica

Table[Sum[Binomial[2 d - 1, d], {d, Divisors[n]}], {n, 1, 26}]
nmax = 26; CoefficientList[Series[Sum[Binomial[2 k - 1, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, binomial(2*d-1,d)); \\ Michel Marcus, Apr 17 2025

Formula

G.f.: Sum_{k>=1} binomial(2*k-1,k) * x^k / (1 - x^k).

a(n) = [(x*y)^n] Sum_{k>=1} x^k / (1 - x^k - y^k).

a(n) = Sum_{d|n} A088218(d).

a(n) = Sum_{d|n} A001700(d-1).

a(n) = Sum_{d|n} A045630(d) * A000005(n/d).

a(n) = A072929(n) / 2.