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%I #32 Nov 24 2023 08:12:51
%S 1,1,0,1,1,1,0,0,2,1,0,0,1,0,0,1,3,2,0,2,1,0,0,2,2,1,0,1,0,0,0,2,4,0,
%T 0,0,1,0,0,0,2,0,0,0,0,1,0,2,3,1,0,2,0,2,0,0,0,2,0,0,1,2,0,0,5,0,0,0,
%U 2,2,0,0,2,0,0,1,2,0,0,2,3,1,0,2,0,2,0,0,0,0,0,0,2,2,0,2,4,0,0,0,1,0,0,0,0
%N Expansion of (phi(q^3)phi(q^5) + phi(q)phi(q^15))/2 in powers of q where phi(q) is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).
%H G. C. Greubel, <a href="/A122855/b122855.txt">Table of n, a(n) for n = 0..5000</a>
%H Alexander Berkovich and Hamza Yesilyurt, <a href="https://doi.org/10.1007/s11139-009-9215-8">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, The Ramanujan Journal, Vol. 20 (2009), pp. 375-408; <a href="https://arxiv.org/abs/math/0611300">arXiv preprint</a>, arXiv:math/0611300 [math.NT], 2006-2007.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of (eta(q^2)^2*eta(q^6)eta(q^10)eta(q^30)^2)/ (eta(q)eta(q^4)eta(q^15)eta(q^60)) in powers of q.
%F a(n) is multiplicative with a(2^e) = |e-1|, a(3^e)=a(5^e)=1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).
%F Euler transform of period 60 sequence [ 1, -1, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 2, 0, 1, -2, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 0, 2, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -1, 1, -2, ...].
%F Moebius transform is period 60 sequence [ 1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, -1, -1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, -1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, -1, 0, ...].
%F a(15n+7) = a(15n+11) = a(15n+13) = a(15n+14) = 0.
%F a(3n) = a(5n) = a(n).
%F G.f.: 1 + Sum_{k>0} Kronecker(-15,k) x^k/(1-(-x)^k).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(15) = 0.811155... . - _Amiram Eldar_, Nov 24 2023
%e 1 + q + q^3 + q^4 + q^5 + 2*q^8 + q^9 + q^12 + q^15 + ...
%t a[0] = 1; a[n_] := DivisorSum[n, KroneckerSymbol[-15, #]*(-1)^Boole[Mod[#, 4] == 2]&]; Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Dec 07 2015, adapted from PARI *)
%o (PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-15,d)*(-1)^(d%4==2)))}
%o (PARI) {a(n)= local(A, p, e); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, e-1, if(p<7, 1, if(p%15==2^valuation(p%15,2), e+1, 1-e%2))))))}
%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^6+A)*eta(x^10+A)*eta(x^30+A)^2/ (eta(x+A)*eta(x^4+A)*eta(x^15+A)*eta(x^60+A)), n))}
%Y A035175(n) = a(4n).
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy,mult
%O 0,9
%A _Michael Somos_, Sep 14 2006
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