%I #13 Mar 17 2024 03:19:59

%S 8,40,128,176,288,640,512,736,1352,1440,1152,2816,1568,2560,4608,3008,

%T 2592,6760,3200,6336,8192,5760,4608,11776,7688,7840,12800,11264,7200,

%U 23040,8192,12160,18432,12960,18432,29744,11552,16000,25088,26496,14112,40960,15488,25344,48672,23040

%N a(n) = sigma(2*n)^2 - sigma(n)^2.

%C Here sigma(n) is the sum of divisors of n (A000203).

%H Paul D. Hanna, <a href="/A227733/b227733.txt">Table of n, a(n) for n = 1..1000</a>

%F Logarithmic derivative of A227732.

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (49/12) * zeta(3) = 4.908399021235... . - _Amiram Eldar_, Mar 17 2024

%e L.g.f.: L(x) = 8*x + 40*x^2/2 + 128*x^3/3 + 176*x^4/4 + 288*x^5/5 + 640*x^6/6 +...

%e where

%e exp(L(x)) = 1 + 8*x + 52*x^2 + 288*x^3 + 1396*x^4 + 6208*x^5 + 25744*x^6 +...+ A227732(n)*x^n +...

%t a[n_] := DivisorSigma[1, 2*n]^2 - DivisorSigma[1, n]^2; Array[a, 50] (* _Amiram Eldar_, Mar 17 2024 *)

%o (PARI) {a(n)=sigma(2*n)^2-sigma(n)^2}

%o for(n=1, 50, print1(a(n), ", "))

%Y Cf. A000203, A002117, A227732.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jul 24 2013