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Expansion of psi(x)^3 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.
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%I #29 Mar 28 2024 09:03:20

%S 1,3,3,4,7,6,9,13,9,10,15,15,13,19,18,16,30,21,19,27,21,31,31,24,25,

%T 39,33,28,48,30,35,54,33,34,52,42,45,51,39,45,55,51,50,70,45,46,78,48,

%U 54,80,57,63,78,54,55,75,84,58,79,60,61,117,63,74,87,72,81

%N Expansion of psi(x)^3 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

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

%H Muniru A Asiru, <a href="/A258835/b258835.txt">Table of n, a(n) for n = 0..20000</a> (first 1000 terms 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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F Expansion of q^(-7/8) * eta(q^2)^6 * eta(q^8)^2 / (eta(q)^3 * eta(q^4)) in powers of q.

%F Euler transform of period 8 sequence [ 3, -3, 3, -2, 3, -3, 3, -4, ...].

%F G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^k)^3 * (1 + x^(2*k)) * (1 + x^(4*k))^2.

%F -8 * a(n) = A121613(4*n + 3). a(n) = sigma(8*n + 7) / 8.

%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/16 = 0.6168502... (A222068). - _Amiram Eldar_, Mar 28 2024

%e G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 7*x^4 + 6*x^5 + 9*x^6 + 13*x^7 + 9*x^8 + ...

%e G.f. = q^7 + 3*q^15 + 3*q^23 + 4*q^31 + 7*q^39 + 6*q^47 + 9*q^55 + 13*q^63 + ...

%p with(numtheory): seq(sigma(8*n-1)/8, n=1..1000); # _Muniru A Asiru_, Dec 31 2017

%t a[ n_] := If[ n < 0, 0, DivisorSigma[ 1, 8 n + 7] / 8];

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^3 EllipticTheta[ 2, 0, x^4] / (16 x^(7/4)), {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, sigma(8*n + 7) / 8)};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^8 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)), n))};

%o (GAP) sequence := List([1..10^5],n->Sigma(8*n-1)/8); # _Muniru A Asiru_, Dec 31 2017

%Y Cf. A000122, A000700, A010054, A121373, A121613, A222068.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Jun 11 2015