login
Expansion of f(x)^6 in powers of x where f() is a Ramanujan theta function.
2

%I #28 Feb 16 2025 08:33:16

%S 1,6,9,-10,-30,0,11,-42,0,70,18,54,49,-90,0,22,-60,0,-110,0,81,-180,

%T -78,0,130,198,0,182,-30,-90,121,-84,0,0,210,0,-252,102,-270,-170,0,0,

%U -69,-330,0,38,420,0,-190,390,0,108,0,0,0,300,99,-442,210,0,418

%N Expansion of f(x)^6 in powers of x where f() is a Ramanujan theta function.

%C Number 59 of the 74 eta-quotients listed in Table I of Martin (1996).

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

%H G. C. Greubel, <a href="/A209941/b209941.txt">Table of n, a(n) for n = 0..2500</a>

%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</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 q^(-1/4) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^6 in powers of q.

%F Euler transform of period 4 sequence [6, -12, 6, -6, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 512 (t/i)^3 f(t) where q = exp(2 Pi i t).

%F a(n) = b(4*n + 1) where b(n) is multiplicative b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 3 (mod 4), else b(p^e) = b(p) * b(p^(e-1)) - p^2 * b^(p^(e-2)) if p == 1 (mod 4).

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

%F a(n) = (-1)^n * A000729(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 9 * a(n).

%F Convolution square of A133089. Convolution cube of A208845. - _Michael Somos_, Jun 09 2015

%e G.f. = 1 + 6*x + 9*x^2 - 10*x^3 - 30*x^4 + 11*x^6 - 42*x^7 + 70*x^9 + ...

%e G.f. = q + 6*q^5 + 9*q^9 - 10*q^13 - 30*q^17 + 11*q^25 - 42*q^29 + 70*q^37 + ...

%p seq(coeff(series(mul((1-(-x)^k)^6,k=1..n), x,n+1),x,n),n=0..70); # _Muniru A Asiru_, Aug 12 2018

%t a[ n_] := SeriesCoefficient[QPochhammer[ x^2]^18 / (QPochhammer[ x] QPochhammer[ x^4])^6, {x, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)

%t CoefficientList[Series[(QPochhammer[x^2]^3/(QPochhammer[x] QPochhammer[x^4]))^6, {x, 0, 50}], x] (* _G. C. Greubel_, Aug 11 2018 *)

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

%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor(4*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==3, if( e%2, 0, p^e), for( i=1, sqrtint(p\2), if( issquare(p - i^2, &y), x=i; break)); a0=1; a1 = y = 2 * (x^2 - y^2) * (-1)^((p%8==5) + y); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))};

%Y Cf. A000729, A133089, A208845.

%K sign,changed

%O 0,2

%A _Michael Somos_, Mar 16 2012