login
A338695
a(n) = Sum_{d|n} 2^(d-1) * binomial(d, n/d).
0
1, 4, 12, 34, 80, 204, 448, 1072, 2308, 5280, 11264, 25088, 53248, 116032, 245920, 527880, 1114112, 2369152, 4980736, 10508880, 22022336, 46193664, 96468992, 201469408, 419430416, 872734720, 1811960832, 3758844096, 7784628224, 16107909312, 33285996544, 68723417856, 141734089728
OFFSET
1,2
FORMULA
G.f.: (1/2) * Sum_{k>=1} ( (2 + 2 * x^k)^k - 2^k ) = Sum_{k>=1} 2^(k-1) * ( (1 + x^k)^k - 1 ).
If p is prime, a(p) = p * 2^(p-1).
MATHEMATICA
a[n_] := DivisorSum[n, 2^(# - 1) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
PROG
(PARI) a(n) = sumdiv(n, d, 2^(d-1)*binomial(d, n/d));
(PARI) N=40; x='x+O('x^N); Vec(sum(k=1, N, (2+2*x^k)^k-2^k)/2)
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 24 2021
STATUS
approved