login
a(n) = Sum_{d|n} d * 4^(d-1).
2

%I #18 Aug 27 2023 17:03:18

%S 1,9,49,265,1281,6201,28673,131337,589873,2622729,11534337,50338105,

%T 218103809,939552777,4026533169,17180000521,73014444033,309238241337,

%U 1305670057985,5497560761865,23089744212017,96757034778633,404620279021569,1688849910733113

%N a(n) = Sum_{d|n} d * 4^(d-1).

%F G.f.: Sum_{k>=1} x^k/(1 - 4 * x^k)^2.

%t a[n_] := DivisorSum[n, 4^(#-1)*# &]; Array[a, 24] (* _Amiram Eldar_, Aug 27 2023 *)

%o (PARI) a(n) = sumdiv(n, d, d*4^(d-1));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-4*x^k)^2))

%Y Cf. A002129, A083413, A359018.

%Y Cf. A167531, A339684, A359190.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Dec 19 2022