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A153728 Expansion of q^(-1/3) * (eta(q)^8 + 8 * eta(q^4)^8) in powers of q^2. 5
1, 20, -70, 56, -125, 308, 110, -520, 57, 0, 182, -880, 1190, 884, 0, -1400, -1330, 1820, -646, 0, -1331, 380, 1120, 2576, 0, 1748, -3850, -3400, 2703, -2500, 3458, 0, -1150, -5236, 0, 6032, 6160, -3220, 4466, 0, -7378, -3920, 0, 2200, 0, 812, -4030, 5600, -4913 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is a member of an infinite family of integer weight modular forms. g_1 = A097195, g_2 = A000727, g_3 = A152243, g_4 = A153728. - Michael Somos, Jun 10 2015

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(3*e/2) if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^3 if p == 1 (mod 6) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x| < |y| and x == 2 (mod 3).

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 648 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A153729.

a(n) = A000731(2*n) = A153729(2*n) = A161969(2*n). - Michael Somos, Jun 10 2015

EXAMPLE

G.f. = 1 + 20*x - 70*x^2 + 56*x^3 - 125*x^4 + 308*x^5 + 110*x^6 - 520*x^7 + ...

G.f. = q + 20*q^7 - 70*q^13 + 56*q^19 - 125*q^25 + 308*q^31 + 110*q^37 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8 + 8 x QPochhammer[ x^4]^8, {x, 0, 2 n}]; (* Michael Somos, Jun 10 2015 *)

PROG

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

(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%6==5, if( e%2, 0, (-p)^(3*e/2)), for(x=1, sqrtint(p\3), if( issquare(p-3*x^2, &y), break)); if( y%3!=1, y=-y); y*=2; y = y^3 - 3*p*y; a0=1; a1=y; for(i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 10 2015 */

(MAGMA) A := Basis( CuspForms( Gamma0(36), 4), 289); A[1] + 20*A[7] - 70*A[12]; /* Michael Somos, Jun 10 2015 */

CROSSREFS

Cf. A000727, A000731, A097195, A152243, A153729, A161969.

Sequence in context: A297597 A186735 A238101 * A234367 A071395 A335557

Adjacent sequences:  A153725 A153726 A153727 * A153729 A153730 A153731

KEYWORD

sign

AUTHOR

Michael Somos, Dec 31 2008

STATUS

approved

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Last modified December 3 14:07 EST 2020. Contains 338906 sequences. (Running on oeis4.)