A030216 - OEIS

%I #32 Mar 12 2021 22:24:42

%S 1,0,0,0,0,-1,0,-1,0,0,-1,0,1,0,-1,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,

%T -1,0,0,2,0,0,0,-1,-1,0,-1,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,

%U 0,0,0,0,1,0,0,1,0,0,0,2,-1,0,0,0,0,0,0,1

%N Expansion of q^-1 * eta(q^10) * eta(q^14) in powers of q^2.

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

%H Seiichi Manyama, <a href="/A030216/b030216.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.

%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 G.f.: Product_{k>=1} (1 - x^(5*k)) * (1 - x^(7*k)). - _Seiichi Manyama_, Oct 18 2016

%F Expansion of f(-x^5) * f(-x^7) in powers of x where f() is a Ramanujan theta function.

%F Euler transform of period 35 sequence [ 0, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -2, ...]. - _Michael Somos_, Oct 19 2016

%e G.f. = 1 - x^5 - x^7 - x^10 + x^12 - x^14 + x^17 + x^19 + x^24 + x^25 - x^32 + ...

%e G.f. = q - q^11 - q^15 - q^21 + q^25 - q^29 + q^35 + q^39 + q^49 + q^51 - q^65 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] QPochhammer[ x^7], {x, 0, n}]; (* _Michael Somos_, Oct 21 2016 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) * eta(x^7 + A), n))}; /* _Michael Somos_, Oct 19 2016 */

%Y Cf. Expansion of eta(q^k)*eta(q^(24 - k)): A030199 (k=1), A030201 (k=3), A030213 (k=5), A030214 (k=7), A030215 (k=9), this sequence (k=10), A030217 (k=11).

%Y Cf. A277582.

%K sign

%O 0,36

%A _N. J. A. Sloane_