%I #23 May 13 2022 14:04:53

%S 1,0,0,5,6,0,0,0,10,0,24,0,0,0,0,21,0,0,40,30,0,0,0,0,31,0,0,0,60,0,

%T 64,0,0,0,0,50,0,0,0,0,84,0,0,120,60,0,0,0,50,0,0,0,0,0,144,0,0,0,120,

%U 0,124,0,0,85,0,0,0,0,0,0,144,0,0,0,0,200,0,0,160,126,91,0,0,0,0,0,0,0

%N Expansion of zeta function of icosian ring.

%C Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187. Then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

%H Amiram Eldar, <a href="/A078473/b078473.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Baake and R. V. Moody, <a href="http://dx.doi.org/10.4153/CJM-1999-057-0">Similarity submodules and root systems in four dimensions</a>, Canad. J. Math. 51 (1999), 1258-1276; <a href="https://arxiv.org/abs/math/9904028">arXiv preprint</a>, arXiv:math/9904028 [math.MG], 1999.

%F Multiplicative with a(p^e) = (5^(e + 1) - 1)/4 if p = 5, (p^(e + 2) - 1)/(p^2 - 1) or 0 if p == 2 or 3 (mod 5) and e is even or odd, respectively, and Sum_{k=0..e} (k + 1)*(e - k + 1)*p^k if p == 1 or 4 (mod 5). - _Amiram Eldar_, May 13 2022

%t f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 13 2022 *)

%o (PARI) {a(n)=local(A); if(n<1, 0, A=direuler(p=2,n,1/(1-X)/(1-kronecker(5,p)*X)); sumdiv(n,d,A[d]*d*A[n/d]))} /* _Michael Somos_, Jun 06 2005 */

%o (PARI) pf(p, r) = {if (p==5, (5^(r+1) -1)/4, if (((p % 5) == 2) || ((p % 5) == 3), if (!(r % 2), (p^(r+2) - 1)/(p^2-1), 0), if (((p % 5) == 1) || ((p % 5) == 4), sum(k=0, r, (k+1)*(r-k+1)*p^k))););}

%o a(n) = {my(f = factor(n)); prod(i=1, #f~, pf(f[i, 1], f[i, 2]));} \\ _Michel Marcus_, Mar 03 2014

%Y Cf. A035187, A035282 (nonzero terms of the sequence), A031363 (n for which a(n) is not zero), A078471.

%K nonn,mult

%O 1,4

%A _Benoit Cloitre_, Dec 31 2002