A229502 - OEIS

%I #27 Sep 08 2022 08:46:05

%S 1,0,-2,-2,0,2,2,4,-1,-2,0,0,-2,-4,2,-4,-2,2,6,4,0,2,-6,-4,3,2,0,4,6,

%T 0,-8,0,-2,0,-4,-2,0,-6,2,-4,0,4,0,-4,2,12,8,8,3,-6,4,0,0,-8,2,0,-6,

%U -6,0,-4,-18,0,2,8,2,0,-10,4,0,4,10,0,4,6,0,0,4

%N Expansion of q * f(-q) * f(-q^4) * f(-q^16) * f(q, -q^3) 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="/A229502/b229502.txt">Table of n, a(n) for n = 1..2500</a> (terms 1..77 from Michael Somos)

%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 Euler transform of period 16 sequence [0, -2, -2, -1, -2, -1, 0, -4, 0, -1, -2, -1, -2, -2, 0, -4, ...].

%F a(2^n) = A090132(n). a(16*n + 5) = a(16*n + 11) = 0. 2 * a(n) = A229893(8*n). a(2*n) = -2 * A229893(n).

%e G.f. = q - 2*q^3 - 2*q^4 + 2*q^6 + 2*q^7 + 4*q^8 - q^9 - 2*q^10 - 2*q^13 + ...

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3, -q^4] QPochhammer[ -q, -q^4] QPochhammer[ -q^4] QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^16], {q, 0, n}];

%o (PARI) {a(n) = my(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, if( issquare( 8*k + 1, &m), (-1)^((m\2 + 2) \ 4) * x^k, 0), A), n))};

%o (Sage) CuspForms( Gamma1(16), 2, prec=78).0;

%o (Magma) Basis( CuspForms( Gamma1(16), 2), 78)[1];

%Y Cf. A090132, A229893.

%K sign

%O 1,3

%A _Michael Somos_, Oct 02 2013