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%I #29 Sep 08 2022 08:46:21
%S 1,7,15,38,40,108,77,188,180,290,187,600,260,560,630,888,442,1323,551,
%T 1620,1218,1364,805,3024,1325,1898,1998,3136,1276,4680,1457,4080,2970,
%U 3230,3290,7470,2072,4028,4134,8200,2542,9072,2795,7656,7830,5888,3337,14496,4998,9825,7038
%N Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.
%C Dirichlet convolution of triangular numbers (A000217) with squares (A000290).
%C a(n) is n times half m, where m is the sum of all parts plus the total number of parts of the partitions of n into equal parts. - _Omar E. Pol_, Nov 30 2019
%H Marius A. Burtea, <a href="/A309732/b309732.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k * (1 + x^k)/(1 - x^k)^3.
%F a(n) = n * (n * d(n) + sigma(n))/2.
%F Dirichlet g.f.: zeta(s-2) * (zeta(s-2) + zeta(s-1))/2.
%F a(n) = n*(A038040(n) + A000203(n))/2 = n*A152211(n)/2. - _Omar E. Pol_, Aug 17 2019
%F a(n) = Sum_{k=1..n} k*sigma(gcd(n,k)). - _Ridouane Oudra_, Nov 28 2019
%p with(numtheory): seq(n*(n*tau(n)+sigma(n))/2, n=1..50); # _Ridouane Oudra_, Nov 28 2019
%t nmax = 51; CoefficientList[Series[Sum[k^2 x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t Table[DirichletConvolve[j (j + 1)/2, j^2, j, n], {n, 1, 51}]
%t Table[n (n DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 51}]
%o (PARI) a(n)=sumdiv(n, d, binomial(n/d+1,2)*d^2); \\ _Andrew Howroyd_, Aug 14 2019
%o (PARI) a(n)=n*(n*numdiv(n) + sigma(n))/2; \\ _Andrew Howroyd_, Aug 14 2019
%o (Magma) [n*(n*NumberOfDivisors(n) + DivisorSigma(1,n))/2:n in [1..51]]; // _Marius A. Burtea_, Nov 29 2019
%Y Cf. A000005, A000203, A000217, A000290, A007437, A034714, A034715, A038040, A064987, A152211, A309731, A244051.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Aug 14 2019
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