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Expansion of phi(q) * phi(q^9) in powers of q where phi() is a Ramanujan theta function.
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%I #44 Feb 16 2025 08:33:25

%S 1,2,0,0,2,0,0,0,0,4,4,0,0,4,0,0,2,0,4,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,

%T 4,0,4,4,0,0,4,0,0,0,0,8,0,0,0,2,0,0,4,0,0,0,0,0,4,0,0,4,0,0,2,0,0,0,

%U 0,0,0,0,4,4,0,0,0,0,0,0,0,4,4,0,0,8,0

%N Expansion of phi(q) * phi(q^9) in powers of q where phi() is a Ramanujan theta function.

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

%H Seiichi Manyama, <a href="/A258034/b258034.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

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

%F Expansion of eta(q^2)^5 * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36))^2 in powers of q.

%F Euler transform of period 36 sequence [2, -3, 2, -1, 2, -3, 2, -1, 4, -3, 2, -1, 2, -3, 2, -1, 2, -6, 2, -1, 2, -3, 2, -1, 2, -3, 4, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).

%F a(n) = (-1)^n * A258322(n). a(4*n) = a(n).

%F a(3*n + 2) = a(4*n + 3) = a(8*n + 6) = a(9*n + 3) = a(9*n + 6) = 0.

%F a(3*n + 1) = 2 * A122865(n). a(6*n + 4) = 2 * A122856(n). a(9*n) = A004018(n). a(12*n + 1) = 2 * A002175(n).

%F a(2*n) = A028601(n). - _Michael Somos_, Jul 04 2015

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - _Amiram Eldar_, Jan 29 2024

%e G.f. = 1 + 2*q + 2*q^4 + 4*q^9 + 4*q^10 + 4*q^13 + 2*q^16 + 4*q^18 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9], {q, 0, n}];

%t a[ n_] := Which[ n < 1, Boole[n == 0], Mod[n, 3] == 2, 0, True, 2 DivisorSum[ n, If[ Mod[n/#, 9] > 0, 1, 2] KroneckerSymbol[ -4, #] &]]; (* _Michael Somos_, Jul 04 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, (n+1)%3 * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A))^2, n))};

%o (PARI) {a(n) = if( n<1, n==0, n%3==2, 0, 2 * sumdiv(n, d, if(n\d%9, 1, 2) * kronecker( -4, d)))}; /* _Michael Somos_, Jul 04 2015 */

%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); (n%3 < 2) * 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)))}; /* _Michael Somos_, Jul 04 2015 */

%o (Magma) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[19];

%Y Cf. A002175, A004018, A019670, A028601, A122856, A122865, A258322.

%Y Cf. A000122, A000700, A010054, A121373.

%K nonn

%O 0,2

%A _Michael Somos_, Jun 03 2015