A309338 a(n) = n^4 if n odd, 7*n^4/8 if n even.

0, 1, 14, 81, 224, 625, 1134, 2401, 3584, 6561, 8750, 14641, 18144, 28561, 33614, 50625, 57344, 83521, 91854, 130321, 140000, 194481, 204974, 279841, 290304, 390625, 399854, 531441, 537824, 707281, 708750, 923521, 917504, 1185921, 1169294, 1500625, 1469664, 1874161, 1824494

Offset

0,3

Comments

Moebius transform of A284900.

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).

Formula

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.

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

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

a(n) = n^4 * (15 - (-1)^n)/16.

a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_4(d).

Sum_{n>=1} 1/a(n) = 113*Pi^4/10080 = 1.091986834012130496797...

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

Mathematica

a[n_] := If[OddQ[n], n^4, 7 n^4/8]; Table[a[n], {n, 0, 38}]
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]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 14, 81, 224, 625, 1134, 2401, 3584, 6561}, 39]
Table[n^4 (15 - (-1)^n)/16, {n, 0, 38}]

Crossrefs

Cf. A000583, A016756, A059377, A129194, A193356, A284900, A309337.

Sequence in context: A099360 A329820 A239421 * A215472 A209942 A215700

Adjacent sequences: A309335 A309336 A309337 * A309339 A309340 A309341

Keyword

nonn,easy,mult

Author

Ilya Gutkovskiy, Jul 24 2019

Status

approved