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%I #15 Nov 10 2020 15:13:09
%S 0,1,7,27,56,125,189,343,448,729,875,1331,1512,2197,2401,3375,3584,
%T 4913,5103,6859,7000,9261,9317,12167,12096,15625,15379,19683,19208,
%U 24389,23625,29791,28672,35937,34391,42875,40824,50653,48013,59319,56000,68921,64827,79507,74536,91125
%N a(n) = n^3 if n odd, 7*n^3/8 if n even.
%C Moebius transform of A007331.
%H Amiram Eldar, <a href="/A309335/b309335.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F G.f.: x * (1 + 7*x + 23*x^2 + 28*x^3 + 23*x^4 + 7*x^5 + x^6)/(1 - x^2)^4.
%F G.f.: Sum_{k>=1} J_3(k) * x^k/(1 - x^(2*k)), where J_3() is the Jordan function (A059376).
%F Dirichlet g.f.: zeta(s-3) * (1 - 1/2^s).
%F a(n) = n^3 * (15 - (-1)^n)/16.
%F a(n) = Sum_{d|n, n/d odd} J_3(d).
%F Sum_{n>=1} 1/a(n) = 57*zeta(3)/56 = 1.223522205001729897639...
%F Multiplicative with a(2^e) = 7*2^(3*e-3), and a(p^e) = p^(3*e) for odd primes p. - _Amiram Eldar_, Oct 26 2020
%F Euler transform is A248882. - _Georg Fischer_, Nov 10 2020
%t a[n_] := If[OddQ[n], n^3, 7 n^3/8]; Table[a[n], {n, 0, 45}]
%t nmax = 45; CoefficientList[Series[x (1 + 7 x + 23 x^2 + 28 x^3 + 23 x^4 + 7 x^5 + x^6)/(1 - x^2)^4, {x, 0, nmax}], x]
%t LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 7, 27, 56, 125, 189, 343}, 46]
%t Table[n^3 (15 - (-1)^n)/16, {n, 0, 45}]
%Y Cf. A000578, A007331, A016755, A026741, A059376, A248882, A303383 (partial sums), A308422, A309336.
%K nonn,easy,mult
%O 0,3
%A _Ilya Gutkovskiy_, Jul 24 2019
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