A226010 - OEIS

%I #19 Sep 08 2022 08:46:05

%S 1,-64,-1836,4096,3990,117504,-433432,-262144,1776573,-255360,1619772,

%T -7520256,-10878466,27739648,-7325640,16777216,60569298,-113700672,

%U -243131740,16343040,795781152,-103665408,-606096456,481296384,-1204783025,696221824,-334611000

%N Expansion of u1 * (u1 + u4) * (u1^2 - 2 * u1 * u4 - u4^2) / eta(q^2)^4 in powers of q where u1 = eta(q)^8 and u4 = 32 * eta(q^4)^8.

%H G. C. Greubel, <a href="/A226010/b226010.txt">Table of n, a(n) for n = 1..2500</a>

%F a(n) is multiplicative with a(2^n) = (-64)^n, a(p^e) = a(p) * a(p^(e-1)) - p^13 * a(p^(e-2)) if p>2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = -128 (t/i)^14 f(t) where q = exp(2 Pi i t).

%F a(2*n) = -64 * a(n).

%e G.f. = q - 64*q^2 - 1836*q^3 + 4096*q^4 + 3990*q^5 + 117504*q^6 - 433432*q^7 + ...

%t eta[q_] := q^(1/24)*QPochhammer[q];  A := eta[q]^8; B := 32*eta[q^4]^8; Drop[CoefficientList[Series[A*(A + B)*(A^2 - 2*A*B - B^2)/eta[q^2]^4, {q, 0, 50}], q], 1] (* _G. C. Greubel_, Aug 09 2018 *)

%o (PARI) {a(n) = my(A, u1, u4); if( n<1, 0, n--; A = x * O(x^n); u1 = eta(x + A)^8; u4 = 32 * x * eta(x^4 + A)^8; polcoeff( u1 * (u1 + u4) * (u1^2 - 2* u1 * u4 - u4^2) / eta(x^2 + A)^4, n))};

%o (Sage) A = CuspForms( Gamma1(2), 14, prec=28) . basis(); A[0] - 64*A[1];

%o (Magma) A := Basis( CuspForms( Gamma1(2), 14), 28); A[1] - 64*A[2];

%Y Cf. A226086.

%K sign,mult

%O 1,2

%A _Michael Somos_, May 22 2013