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A279371 Expansion of F(q) + 4*F(q^2) + 8*F(q^4) in powers of q where F(q) = q * (f(-q) * f(-q^11))^2. 2

%I #15 Sep 08 2022 08:46:18

%S 1,2,-1,2,1,-2,-2,-8,-2,2,1,-2,4,-4,-1,12,-2,-4,0,2,2,2,-1,8,-4,8,5,

%T -4,0,-2,7,-8,-1,-4,-2,-4,3,0,-4,-8,-8,4,-6,2,-2,-2,8,-12,-3,-8,2,8,

%U -6,10,1,16,0,0,5,-2,12,14,4,-8,4,-2,-7,-4,1,-4,-3,16,4

%N Expansion of F(q) + 4*F(q^2) + 8*F(q^4) in powers of q where F(q) = q * (f(-q) * f(-q^11))^2.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Fourier expansion of a multiplicative weight 2 cusp form on Gamma_0(44).

%H G. C. Greubel, <a href="/A279371/b279371.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) for p != 11.

%F a(2*n + 1) = A006571(2*n + 1).

%e G.f. = q + 2*q^2 - q^3 + 2*q^4 + q^5 - 2*q^6 - 2*q^7 - 8*q^8 - 2*q^9 + ...

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2 + 4 q^2 (QPochhammer[ q^2] QPochhammer[ q^22])^2 + 8 q^4 (QPochhammer[ q^4] QPochhammer[ q^44])^2, {q, 0, n}];

%o (PARI) {a(n) = my(A, F); if( n<1, 0, A = x * O(x^n); F = x * (eta(x + A) * eta(x^11 + A))^2; polcoeff( F + 4*subst(F, x, x^2) + 8*subst(F, x, x^4), n))};

%o (Magma) A := Basis( CuspForms( Gamma0(44), 2), 79); A[1] + 2*A[2] - A[3] + 2*A[4];

%Y Cf. A006571.

%K sign,mult

%O 1,2

%A _Michael Somos_, Dec 10 2016

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)