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Euler transform of sigma_n(n) (sum of n-th powers of divisors of n).
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%I #6 Apr 28 2023 20:15:55

%S 1,1,6,34,322,3588,52844,900082,18111465,411941506,10548286788,

%T 298667744593,9286665651198,314077164671106,11484692279345752,

%U 451291302965764596,18966834595501974235,848853415894558707472,40305029983754331855502,2023571200162099967806430,107109031661019664234558776

%N Euler transform of sigma_n(n) (sum of n-th powers of divisors of n).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerTransform.html">Euler Transform</a>

%F G.f.: Product_{k>=1} 1/(1 - x^k)^sigma(k,k).

%F a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k) for n>1, a(0)=1, where b(k) = Sum_{d|k} d*sigma(d,d).

%t a = Table[DivisorSigma[n, n], {n, 20}]; CoefficientList[Series[Product[1/(1 - x^m)^a[[m]], {m, 20}], {x, 0, 20}], x]

%Y Cf. A023887 (sigma_n(n)), A061256, A350503, A353233.

%K nonn,easy

%O 0,3

%A _Wesley Ivan Hurt_, Apr 28 2023