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A122266 Expansion of f(-q)^2 * Q(q) in powers of q. 2
1, 238, 1679, 2162, 2401, -6958, -1442, -23040, 1918, -9362, 14641, 0, 80640, -20398, 28083, 64078, -38398, -69120, 0, -90482, -58562, 0, -241920, 100558, 146879, 0, -193438, 399602, 104638, 114002, 130321, 24242, 0, 107282, -276962, 351118 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

f(-q) (g.f. A010815) and Q(q) (g.f. A004009) are Ramanujan q-series.

LINKS

Table of n, a(n) for n=0..35.

S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).

FORMULA

Expansion of q^(-1/12) * (eta(q)^16 + 256 * eta(q)^8 * eta(q^4)^8 + 4096 * eta(q^4)^16) * eta(q)^2 / eta(q^2)^8 in powers of q. - Michael Somos, Jun 25 2013

a(n) = b(12*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 7 or 11 (mod 12), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 or 5 (mod 12). - Michael Somos, Jun 24 2013

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12^5 (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 06 2014

Convolution of A002107 and A004009. - Michael Somos, Jun 25 2013

EXAMPLE

G.f. = 1 + 238*x + 1679*x^2 + 2162*x^3 + 2401*x^4 - 6958*x^5 - 1442*x^6 + ...

G.f. = q + 238*q^13 + 1679*q^25 + 2162*q^37 + 2401*q^49 - 6958*q^61 - 1442*q^73 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n}]) QPochhammer[ q]^2, {q, 0, n}] (* Michael Somos, Jun 24 2013 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 (EllipticTheta[ 4, 0, q]^8 + EllipticTheta[ 2, 0, q^(1/2)]^8), {q, 0, n}] (* Michael Somos, Jun 25 2013 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * sum( k=1, n, 240 * sigma( k, 3) * x^k, 1 + A), n))}

(PARI) {a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = 64 * x * (eta(x^4 + A) / eta(x + A))^8; polcoeff( (1 + 4*B + B^2) * eta(x + A)^18 / eta(x^2 + A)^8, n))} /* Michael Somos, Jun 25 2013 */

(PARI) {a(n) = local(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 12*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p<5, 0, if( p%12 > 6, if( e%2, 0, p^(2*e)), forstep( i = 1, sqrtint( p), 2, if( issquare( p - i^2, &y), x=i; break)); if( p%12 == 5, a1 = 8 * x*y * (x-y) * (x+y) * (-1)^((x%6==1) + (y%6==4)), a1 = 2 * (x^2-y^2+2*x*y) * (x^2-y^2-2*x*y) * (-1)^(x%6==3) ); a0 = 1; y = a1; for( i=2, e, x = y * a1 - p^4 * a0; a0=a1; a1=x); a1 )))))} /* Michael Somos, Jun 24 2013 */

CROSSREFS

Cf. A002107, A004009, A010818.

Sequence in context: A264885 A252910 A252905 * A286213 A251404 A238602

Adjacent sequences:  A122263 A122264 A122265 * A122267 A122268 A122269

KEYWORD

sign

AUTHOR

Michael Somos, Aug 28 2006

STATUS

approved

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Last modified October 20 13:38 EDT 2021. Contains 348108 sequences. (Running on oeis4.)