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%I #11 Oct 26 2020 16:33:08
%S 0,1,14,81,224,625,1134,2401,3584,6561,8750,14641,18144,28561,33614,
%T 50625,57344,83521,91854,130321,140000,194481,204974,279841,290304,
%U 390625,399854,531441,537824,707281,708750,923521,917504,1185921,1169294,1500625,1469664,1874161,1824494
%N a(n) = n^4 if n odd, 7*n^4/8 if n even.
%C Moebius transform of A284900.
%H Amiram Eldar, <a href="/A309338/b309338.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-10,0,10,0,-5,0,1).
%F G.f.: x * (1 + 14*x + 76*x^2 + 154*x^3 + 230*x^4 + 154*x^5 + 76*x^6 + 14*x^7 + x^8)/(1 - x^2)^5.
%F G.f.: Sum_{k>=1} J_4(k) * x^k/(1 + x^k), where J_4() is the Jordan function (A059377).
%F Dirichlet g.f.: zeta(s-4) * (1 - 2^(1-s)).
%F a(n) = n^4 * (15 - (-1)^n)/16.
%F a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_4(d).
%F Sum_{n>=1} 1/a(n) = 113*Pi^4/10080 = 1.091986834012130496797...
%F Multiplicative with a(2^e) = 7*2^(4*e-3), and a(p^e) = p^(4*e) for odd primes p. - _Amiram Eldar_, Oct 26 2020
%t a[n_] := If[OddQ[n], n^4, 7 n^4/8]; Table[a[n], {n, 0, 38}]
%t nmax = 38; CoefficientList[Series[x (1 + 14 x + 76 x^2 + 154 x^3 + 230 x^4 + 154 x^5 + 76 x^6 + 14 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
%t LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 14, 81, 224, 625, 1134, 2401, 3584, 6561}, 39]
%t Table[n^4 (15 - (-1)^n)/16, {n, 0, 38}]
%Y Cf. A000583, A016756, A059377, A129194, A193356, A284900, A309337.
%K nonn,easy,mult
%O 0,3
%A _Ilya Gutkovskiy_, Jul 24 2019
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