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%I #10 Jul 23 2022 03:27:41
%S 1,6,15,36,56,111,141,240,300,446,507,791,820,1161,1310,1736,1786,
%T 2505,2471,3346,3466,4307,4325,5895,5581,7026,7230,8905,8556,11246,
%U 10417,13176,13050,15476,15106,19391,17576,21495,21374,25690,23822,30162,27435,33707,32990,37841,35721
%N Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.
%C Inverse Möbius transform of square pyramidal numbers (A000330).
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: Sum_{k>=1} A000330(k)*x^k/(1 - x^k).
%F a(n) = Sum_{d|n} d*(d + 1)*(2*d + 1)/6.
%F a(n) = (A000203(n) + 3*A001157(n) + 2*A001158(n))/6.
%F a(n) = Sum_{i=1..n} i^2*A135539(n,i). - _Ridouane Oudra_, Jul 22 2022
%p a:=series(add(x^k*(1+x^k)/(1-x^k)^4,k=1..100),x=0,48): seq(coeff(a,x,n),n=1..47); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 47; Rest[CoefficientList[Series[Sum[x^k (1 + x^k)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
%t Table[Sum[d (d + 1) (2 d + 1)/6, {d, Divisors[n]}], {n, 47}]
%t Table[(DivisorSigma[1, n] + 3 DivisorSigma[2, n] + 2 DivisorSigma[3, n])/6, {n, 47}]
%Y Cf. A000203, A000330, A001157, A001158, A007437, A059358.
%Y Cf. A135539.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Oct 24 2018
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