cp's OEIS Frontend

A351405 a(1) = 1; a(n+1) = Sum_{d|n} 2^(n/d - 1) * a(d).

%I A351405 #7 Feb 18 2022 12:44:14
%S A351405 1,1,3,7,17,33,75,139,289,557,1119,2143,4341,8437,16843,33343,66573,
%T A351405 132109,264243,526387,1052549,2101617,4202031,8396335,16792705,
%U A351405 33570193,67137403,134248191,268492033,536927489,1073853307,2147595131,4295180241,8590155085
%N A351405 a(1) = 1; a(n+1) = Sum_{d|n} 2^(n/d - 1) * a(d).
%F A351405 G.f. A(x) satisfies: A(x) = x * ( 1 + A(x) + 2 * A(x^2) + 4 * A(x^3) + ... + 2^(k-1) * A(x^k) + ... ).
%F A351405 G.f.: x * ( 1 + Sum_{n>=1} a(n) * x^n / (1 - 2 * x^n) ).
%F A351405 a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, Feb 18 2022
%p A351405 a:= proc(n) option remember; `if`(n=1, 1,
%p A351405       add(2^((n-1)/d-1)*a(d), d=numtheory[divisors](n-1)))
%p A351405     end:
%p A351405 seq(a(n), n=1..34);  # _Alois P. Heinz_, Feb 10 2022
%t A351405 a[1] = 1; a[n_] := a[n] = Sum[2^((n - 1)/d - 1) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 34}]
%t A351405 nmax = 34; A[_] = 0; Do[A[x_] = x (1 + Sum[2^(k - 1) A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%Y A351405 Cf. A003238, A038046, A054599, A308076.
%K A351405 nonn
%O A351405 1,3
%A A351405 _Ilya Gutkovskiy_, Feb 10 2022