A209941 Expansion of f(x)^6 in powers of x where f() is a Ramanujan theta function.

1, 6, 9, -10, -30, 0, 11, -42, 0, 70, 18, 54, 49, -90, 0, 22, -60, 0, -110, 0, 81, -180, -78, 0, 130, 198, 0, 182, -30, -90, 121, -84, 0, 0, 210, 0, -252, 102, -270, -170, 0, 0, -69, -330, 0, 38, 420, 0, -190, 390, 0, 108, 0, 0, 0, 300, 99, -442, 210, 0, 418

Offset

0,2

Comments

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/4) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^6 in powers of q.

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

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).

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).

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

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

Convolution square of A133089. Convolution cube of A208845. - Michael Somos, Jun 09 2015

Example

G.f. = 1 + 6*x + 9*x^2 - 10*x^3 - 30*x^4 + 11*x^6 - 42*x^7 + 70*x^9 + ...
G.f. = q + 6*q^5 + 9*q^9 - 10*q^13 - 30*q^17 + 11*q^25 - 42*q^29 + 70*q^37 + ...

Maple

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

Mathematica

a[ n_] := SeriesCoefficient[QPochhammer[ x^2]^18 / (QPochhammer[ x] QPochhammer[ x^4])^6, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
CoefficientList[Series[(QPochhammer[x^2]^3/(QPochhammer[x] QPochhammer[x^4]))^6, {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

Prog

(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))};
(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)))};

Crossrefs

Cf. A000729, A133089, A208845.

Sequence in context: A328234 A051221 A029843 * A000729 A280666 A282937

Adjacent sequences: A209938 A209939 A209940 * A209942 A209943 A209944

Keyword

sign

Author

Michael Somos, Mar 16 2012

Status

approved