login
Expansion of phi(-x) * f(-x)^2 * f(-x^6)^3 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
2

%I #10 Feb 16 2025 08:33:39

%S 1,-4,3,4,-1,-4,-11,20,-9,-8,8,12,16,-28,-3,-16,-7,0,37,-12,32,40,-37,

%T -32,-65,44,-16,-8,41,48,-34,8,-39,4,64,-40,24,-4,3,-32,-33,-12,43,

%U -88,81,-28,-67,44,103,32,50,88,-79,-48,-80,8,-176,-68,53,132,-16

%N Expansion of phi(-x) * f(-x)^2 * f(-x^6)^3 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

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

%H Seiichi Manyama, <a href="/A280822/b280822.txt">Table of n, a(n) for n = 0..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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of phi(-x)^3 * f(-x^6)^3 / chi(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%F Expansion of q^(-5/6) * eta(q)^4 * eta(q^6)^3 / eta(q^2) in powers of q.

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

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

%F 6 * a(n) = A116418(4*n + 3).

%e G.f. = 1 - 4*x + 3*x^2 + 4*x^3 - x^4 - 4*x^5 - 11*x^6 + 20*x^7 - 9*x^8 + ...

%e G.f. = q^5 - 4*q^11 + 3*q^17 + 4*q^23 - q^29 - 4*q^35 - 11*q^41 + 20*q^47 + ...

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

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

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

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^4 * (1 - x^(2*k))^2 * (1 + x^(2*k) + x^(4*k))^3, 1 + x * O(x^n)), n))};

%Y Cf. A116418.

%K sign,changed

%O 0,2

%A _Michael Somos_, Jan 08 2017