A227618 - OEIS

%I #13 Sep 08 2022 08:46:05

%S 1,-1,-1,0,-1,2,0,2,0,0,1,-2,-1,-2,2,-4,-2,1,3,2,2,0,0,0,0,2,-4,4,3,

%T -2,-8,4,-2,2,-2,-2,3,-6,0,-4,0,0,5,-2,1,6,8,4,3,-3,2,-2,-5,0,0,-4,0,

%U -6,-3,0,-9,8,2,0,2,2,-5,0,-6,4,0,2,0,0,3,6,2,-2

%N Expansion of q * f(-q) * f(-q^4) * f(-q^16) * f(-q^6, -q^10) in powers of q where f() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A227618/b227618.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q * f(-q) * f(-q^2) * f(-q^16)^3 / f(-q^2, -q^14) in powers of q where f() is a Ramanujan theta function.

%F Expansion of q * psi(-q) * psi(-q^2) * psi(-q^4) * f(-q^6, -q^10) in powers of q where psi(), f() are Ramanujan theta functions.

%F Euler transform of period 16 sequence [ -1, -1, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -1, -1, -4, ...].

%e G.f. = q - q^2 - q^3 - q^5 + 2*q^6 + 2*q^8 + q^11 - 2*q^12 - q^13 - 2*q^14 + ...

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^6, q^16] QPochhammer[ q^10, q^16] QPochhammer[ q^16]^2 QPochhammer[ q] QPochhammer[ q^4], {q, 0, n}];

%o (PARI) {a(n) = local(A, m); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^16 + A) * sum( k=0, n\2, if( issquare( 16*k + 1, &m), (-1)^((m + 1) \ 8) * x^(2*k), 0), A), n))};

%o (Sage) A = CuspForms( Gamma1(16), 2, prec=80).basis(); A[0] - A[1];

%o (Magma) A := Basis( CuspForms( Gamma1(16), 2), 80); A[1] - A[2]; /* _Michael Somos_, Jan 08 2015 */

%K sign

%O 1,6

%A _Michael Somos_, Jul 17 2013