A115155 - OEIS

%I #36 Jun 14 2023 09:32:18

%S 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,

%T -27,0,0,-15,2,33,0,-14,0,-27,0,-22,0,-35,0,0,0,0,45,34,-14,-15,49,25,

%U 42,0,-86,-27,0,0,66,0,0,45,-118,2,0,13,0,0,0,42,-102,0,0

%N Expansion of (eta(q^3) * eta(q^5))^3 + (eta(q) * eta(q^15))^3 in powers of q.

%C The terms of A136549 differ only in sign from this sequence. - _Michael Somos_, Jun 14 2023

%H G. C. Greubel, <a href="/A115155/b115155.txt">Table of n, a(n) for n = 1..1000</a>

%H Steven R. Finch, <a href="/A000521/a000521_1.pdf">Modular forms on SL_2(Z)</a>, December 28, 2005, see page 5. [Cached copy, with permission of the author]

%H M. D. Rogers, <a href="http://arxiv.org/abs/0806.3590">Hypergeometric formulas for lattice sums and Mahler measures</a>. see p. 26 f(q)

%F a(n) is multiplicative with a(3^e) = (-3)^e, a(5^e) = 5^e, a(p^e) = p^e if e even else 0 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).

%F 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_, Jun 23 2012

%F a(2^n) = A106853(n).

%F a(n) = A030220(n) + A136599(n). - _Michael Somos_, Oct 13 2015

%e 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 +...

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] QPochhammer[ q^5])^3 + q^2 (QPochhammer[ q] QPochhammer[ q^15])^3, {q, 0, n}]; (* _Michael Somos_, May 24 2013 *)

%o (PARI) {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))};

%o (Sage) A = CuspForms( Gamma1(15), 3, prec=80) . basis(); A[0] + A[1] - 3*A[2] - 3*A[3] + 5*A[4] - 3*A[5] - 7*A[7]; # _Michael Somos_, May 28 2013

%o (Magma) 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 */

%Y Cf. A030220, A106853, A136549, A136599.

%K sign,mult

%O 1,3

%A _Michael Somos_, Jan 14 2006