A316662 - OEIS

%I #11 Sep 08 2022 08:46:22

%S 1,1,1,3,3,4,3,4,6,4,7,7,7,9,10,7,7,9,12,12,9,16,13,15,13,12,13,16,19,

%T 13,18,16,19,21,19,20,21,21,24,19,22,24,22,27,25,21,21,27,30,32,27,28,

%U 30,37,25,31,34,28,36,28,31,28,31,48,36,36,38,31,42,37

%N Expansion of f(x, x^2) * psi(x^3)^3 in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

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

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A316662/b316662.txt">Table of n, a(n) for n = 0..2500</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 Expansion of q^(-7/6) * eta(q^2) * eta(q^6)^5 / (eta(q) * eta(q^3)) in powers of q.

%F Expansion of b(x^2) * c(x^2)^2 / sqrt(b(x) * c(x)/3) in powers of x where b(), c() are cubic AGM theta functions.

%F Expansion of psi(x^3) * f(-x^6)^3 / chi(-x) = f(-x^6)^4 / (chi(-x) * chi(-x^3)) =  f(-x^6)^5 /(f(-x^3) * chi(-x)) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

%F G.f.: Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * (1 - x^(6*k))^4.

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

%e G.f. = q^7 + q^13 + q^19 + 3*q^25 + 3*q^31 + 4*q^37 + 3*q^43 + 4*q^49 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^4 / (QPochhammer[ x, x^2] QPochhammer[ x^3, x^6]), {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^5 / (QPochhammer[ x, x^2] QPochhammer[ x^3]), {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^6]^5 / (QPochhammer[ x] QPochhammer[x^3]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)), n))};

%o (Magma) Basis( ModularForms( Gamma0(36), 2), 427) [8];

%K nonn

%O 0,4

%A _Michael Somos_, Jul 09 2018