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%I #18 Dec 16 2023 09:01:07
%S 1,1,0,0,1,2,0,0,2,2,0,0,1,1,0,0,2,0,0,0,2,2,0,0,1,2,0,0,2,2,0,0,0,2,
%T 0,0,2,1,0,0,1,2,0,0,2,0,0,0,2,2,0,0,0,2,0,0,2,0,0,0,3,2,0,0,2,2,0,0,
%U 2,2,0,0,0,1,0,0,2,0,0,0,0,2,0,0,1,2,0,0,2,2,0,0,0,4,0,0,2,0,0,0,2,0,0,0,4
%N Sum over divisors d of 2n+1 of Kronecker(-18/d).
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.58).
%H Antti Karttunen, <a href="/A122071/b122071.txt">Table of n, a(n) for n = 0..10000</a>
%F Expansion of q^(-1/2)*eta(q^2)^2*eta(q^3)*eta(q^8)*eta(q^12)^3/ (eta(q)*eta(q^4)^2*eta(q^6)*eta(q^24)) in powers of q.
%F Euler transform of period 24 sequence [1, -1, 0, 1, 1, -1, 1, 0, 0, -1, 1, -2, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 1, -2, ...].
%F a(n) = b(2n+1) where b(n) is multiplicative and b(2^e)=0^e, b(3^e)=1, b(p^e) = e+1 if p == 1,3 (mod 8), b(p^e) = (1+(-1)^e)/2 if p == 5,7 (mod 8).
%F G.f.: Sum_{k>0} x^k(1-x^(4k-2))(1-x^(6k-3))/(1+x^(12k-6)).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/(3*sqrt(2)) = 0.7404804... (A093825). - _Amiram Eldar_, Dec 16 2023
%t a[n_] := DivisorSum[2*n+1, KroneckerSymbol[-18, #] &]; Array[a, 100, 0] (* _Amiram Eldar_, Dec 16 2023 *)
%o (PARI) {a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-18,d)))}
%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^3+A)*eta(x^8+A)*eta(x^12+A)^3/ eta(x+A)/eta(x^4+A)^2/eta(x^6+A)/eta(x^24+A), n))}
%o (PARI) {a(n)=local(A, p, e); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==3, 1, if(p%8<4, e+1, (1+(-1)^e)/2))))))}
%o (PARI) a(n) = sumdiv(2*n+1, d, kronecker(-18, d)); \\ _Michel Marcus_, Jul 28 2017
%Y A035172(2n+1) = a(n).
%Y Cf. A093825.
%K nonn,easy
%O 0,6
%A _Michael Somos_, Aug 20 2006
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