A360802 - OEIS

%I #14 Aug 02 2023 02:00:15

%S 1,3,5,17,17,105,65,449,641,1953,1025,16257,4097,37761,93185,247809,

%T 65537,1499649,262145,6596609,8847361,13654017,4194305,210026497,

%U 90177537,251764737,833880065,2659418113,268435457,18345328641,1073741825,53553922049,75438751745

%N Expansion of Sum_{k>0} (x / (1 - (2 * x)^k))^k.

%F a(n) = Sum_{d|n} 2^(n-d) * binomial(d+n/d-2,d-1).

%F If p is prime, a(p) = 1 + 2^(p-1).

%t a[n_] := DivisorSum[n, 2^(n-#) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* _Amiram Eldar_, Aug 02 2023 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(2*x)^k))^k))

%o (PARI) a(n) = sumdiv(n, d, 2^(n-d)*binomial(d+n/d-2, d-1));

%Y Cf. A157019, A339481.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Feb 21 2023