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A259761
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Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function.
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4
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1, 2, 2, 0, 2, 4, 0, 0, 2, 4, 4, 0, 0, 4, 0, 0, 2, 4, 4, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 4, 0, 0, 2, 0, 4, 0, 4, 4, 0, 0, 4, 4, 0, 0, 0, 8, 0, 0, 0, 2, 6, 0, 4, 4, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 4, 4, 4, 0, 0, 8, 0
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of phi(x) * phi(x^9) + 2 * x^2 * chi(x^3)^2 * psi(-x^9)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
a(n) = 2 * b(n) with a(0) = 1 and b() is multiplicative with b(2^e) = 1, b(3^e) = 1 + (-1)^e if e>0, b(p^e) = e+1 if p == 1, 5 (mod 12), (p^e) = (1 + (-1)^e)/2 if p == 7, 11 (mod 12).
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5*Pi/9 = 1.745329... (= 100 * A019685). - Amiram Eldar, Dec 29 2023
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EXAMPLE
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G.f. = 1 + 2*x + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^8 + 4*x^9 + 4*x^10 + 4*x^13 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^2 + EllipticTheta[ 3, 0, x^9]^2) / 2, {x, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ ( Which[ # < 3, 1, # == 3, 1 + (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, (1 + (-1)^#2) / 2 ] & @@@ FactorInteger[n])];
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PROG
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(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))};
(Magma) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[3] + 2*A[5] + 4*A[6] + 2*A[9] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[18] + 4*A[19];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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