A262064 - OEIS

%I #9 Mar 12 2021 22:24:48

%S 1,0,1,0,2,0,3,0,5,1,7,1,11,2,15,4,22,6,30,9,42,14,56,20,77,29,101,41,

%T 135,57,176,78,231,107,297,143,385,191,490,253,627,332,793,432,1003,

%U 561,1257,721,1578,924,1963,1177,2443,1492,3022,1882,3734,2367,4589

%N Expansion of f(x^9, x^15) / f(-x^2, -x^4) in powers of x where f(, ) is the Ramanujan general 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="/A262064/b262064.txt">Table of n, a(n) for n = 0..2500</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 Euler transform of period 48 sequence [ 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, ...].

%F a(n) = A143067(2*n).

%e G.f. = 1 + x^2 + 2*x^4 + 3*x^6 + 5*x^8 + x^9 + 7*x^10 + x^11 + 11*x^12 + ...

%e G.f. = q^5 + q^101 + 2*q^197 + 3*q^293 + 5*q^389 + q^437 + 7*q^485 + q^533 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^9, x^24] QPochhammer[ -x^15, x^24] QPochhammer[ x^24] / QPochhammer[ x^2], {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( subst( prod(k=1, n\3, 1 - x^k * [1, 0, 0, 1, 0, 1, 0, 0][k%8 + 1], 1 + x * O(x^(n\3))), x, -x^3) / eta(x^2 + x * O(x^n)), n))};

%Y Cf. A143067.

%K nonn

%O 0,5

%A _Michael Somos_, Sep 10 2015