A115155 -id:A115155 - cp's OEIS frontend

A136549 Expansion of (eta(q^3) * eta(q^5))^3 - (eta(q) * eta(q^15))^3 in powers of q.

1, -1, 3, -3, -5, -3, 0, 7, 9, 5, 0, -9, 0, 0, -15, 5, 14, -9, -22, 15, 0, 0, -34, 21, 25, 0, 27, 0, 0, 15, 2, -33, 0, -14, 0, -27, 0, 22, 0, -35, 0, 0, 0, 0, -45, 34, 14, 15, 49, -25, 42, 0, 86, -27, 0, 0, -66, 0, 0, 45, -118, -2, 0, 13, 0, 0, 0, -42, -102, 0, 0, 63, 0, 0, 75, 66, 0, 0, 98, -25, 81, 0, -154, 0

Offset: 1

Michael Somos, Jan 05 2008

G.f. is a newform level 15 weight 3 and nontrivial character.

The terms of A115155 differ only in sign from this sequence. - Michael Somos, Jun 14 2023

			G.f. = q - q^2 + 3*q^3 - 3*q^4 - 5*q^5 - 3*q^6 + 7*q^8 + 9*q^9 + 5*q^10 + ...

Cf. A115155.

A := Basis( CuspForms( Gamma1(15), 3), 80); A[1] - A[2] + 3*A[3] - 3*A[4] - 5*A[5] - 3*A[6] + 7*A[8]; /* Michael Somos, Oct 13 2015 */

QP = QPochhammer; s = (QP[q^3]*QP[q^5])^3-q*(QP[q]*QP[q^15])^3 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A))^3 - x * (eta(x + A) * eta(x^15 + A))^3, n))};

a(n) is multiplicative with a(3^e) = 3^e, a(5^e) = (-5)^e, a(p^e) = p^e * (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = a(p) * a(p^(e-1)) - p^2 * a(p^(e-2)) if p == 1, 2, 4, 8 (mod 15).

G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 29 2013

A136599 Expansion of (eta(q) * eta(q^15))^3 in powers of q.

Original entry on oeis.org

1, -3, 0, 5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, -14, 9, 0, -15, 0, 0, 34, 0, 0, 0, -27, 0, 0, -15, 0, 33, 0, 0, 0, 0, 0, -22, 0, 0, 0, 0, 0, 0, 45, 0, -14, -15, 0, 25, 0, 0, -86, 0, 0, 0, 66, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 42, 0, 0, 0, -63, 0, 0, -75, 0, 0, 0, 0

Offset: 2

Michael Somos, Jan 11 2008

sign

			G.f. = q^2 - 3*q^3 + 5*q^5 - 7*q^8 + 9*q^12 - 14*q^17 + 9*q^18 - 15*q^20 + ...

Cf. A028955, A030220, A115155.

A := Basis( CuspForms( Gamma1(15), 3), 80); A[2] - 3*A[3] + 5*A[5] - 7*A[8]; /* Michael Somos, Oct 13 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^15])^3, {q, 0, n}]; (* Michael Somos, Oct 13 2015 *)

{a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^15 + A))^3, n))};

Euler transform of period 15 sequence [ -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
a(n) nonzero or n=0 if and only if n is in A028955.
G.f.: x^2 * (Product_{k>0} (1 - x^k) * (1 - x^(15*k)))^3.
a(3*n) = -3 * A030220(n). a(3*n + 1) = 0. - Michael Somos, Oct 13 2015
A115155(n) = a(n) + A030220(n). - Michael Somos, Oct 13 2015

cp's OEIS Frontend

A136549 Expansion of (eta(q^3) * eta(q^5))^3 - (eta(q) * eta(q^15))^3 in powers of q.

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