A309336 - OEIS

%I #11 Oct 26 2020 16:33:01

%S 0,1,15,81,240,625,1215,2401,3840,6561,9375,14641,19440,28561,36015,

%T 50625,61440,83521,98415,130321,150000,194481,219615,279841,311040,

%U 390625,428415,531441,576240,707281,759375,923521,983040,1185921,1252815,1500625,1574640,1874161,1954815

%N a(n) = n^4 if n odd, 15*n^4/16 if n even.

%C Moebius transform of A285989.

%H Amiram Eldar, <a href="/A309336/b309336.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 + 15*x + 76*x^2 + 165*x^3 + 230*x^4 + 165*x^5 + 76*x^6 + 15*x^7 + x^8)/(1 - x^2)^5.

%F G.f.: Sum_{k>=1} J_4(k) * x^k/(1 - x^(2*k)), where J_4() is the Jordan function (A059377).

%F Dirichlet g.f.: zeta(s-4) * (1 - 1/2^s).

%F a(n) = n^4 * (31 - (-1)^n)/32.

%F a(n) = Sum_{d|n, n/d odd} J_4(d).

%F Sum_{n>=1} 1/a(n) = 241*Pi^4/21600 = 1.086832913851601267313987...

%F Multiplicative with a(2^e) = 15*2^(4*e-4), and a(p^e) = p^(4*e) for odd primes p. - _Amiram Eldar_, Oct 26 2020

%t a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]

%t nmax = 38; CoefficientList[Series[x (1 + 15 x + 76 x^2 + 165 x^3 + 230 x^4 + 165 x^5 + 76 x^6 + 15 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, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]

%t Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]

%Y Cf. A000583, A016756, A026741, A059377, A285989, A308422, A309335.

%K nonn,easy,mult

%O 0,3

%A _Ilya Gutkovskiy_, Jul 24 2019