A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561, 9375, 14641, 19440, 28561, 36015, 50625, 61440, 83521, 98415, 130321, 150000, 194481, 219615, 279841, 311040, 390625, 428415, 531441, 576240, 707281, 759375, 923521, 983040, 1185921, 1252815, 1500625, 1574640, 1874161, 1954815

Offset: 0

Author

Ilya Gutkovskiy, Jul 24 2019

Keywords

• nonn
• easy
• mult

Comments

Moebius transform of A285989.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Crossrefs

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

Programs

Mathematica

a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]
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]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]
Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]

Formula

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.

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

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

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

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

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

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