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A137677 Expansion of f(-x, -x^4) / psi(-x) where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

%I A137677 #12 Feb 16 2025 08:33:07
%S A137677 1,0,0,1,0,0,0,1,1,0,0,1,1,0,0,2,2,0,0,2,2,0,0,3,4,0,0,4,4,0,0,5,6,0,
%T A137677 0,7,7,0,0,9,10,0,0,11,11,0,0,14,16,0,0,18,18,0,0,22,24,0,0,27,28,0,0,
%U A137677 34,36,0,0,41,42,0,0,50,54,0,0,61,62,0,0,73
%N A137677 Expansion of f(-x, -x^4) / psi(-x) where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
%C A137677 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A137677 G. C. Greubel, <a href="/A137677/b137677.txt">Table of n, a(n) for n = 0..1000</a>
%H A137677 G. E. Andrews, <a href="http://dx.doi.org/10.1090/cbms/066">q-series</a>, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 36, Equ. (4.12). MR0858826 (88b:11063).
%H A137677 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A137677 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A137677 Expansion of f(-x^2) * f(-x^5) / ( f(-x^4) * f(-x^2, -x^3) ) in powers of x where f(, ) is Ramanujan's general theta function.
%F A137677 Expansion of ( f(-x^11, -x^19) + x^3 * f(-x, -x^29) ) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.
%F A137677 Euler transform of period 20 sequence [ 0, 0, 1, 0, 0, -1, 1, 1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, 0, ...].
%F A137677 a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A122134(n). a(4*n + 3) = A122130(n).
%e A137677 G.f. = 1 + x^3 + x^7 + x^8 + x^11 + x^12 + 2*x^15 + 2*x^16 + 2*x^19 + 2*x^20 + ...
%e A137677 G.f. = q + q^31 + q^71 + q^81 + q^111 + q^121 + 2*q^151 + 2*q^161 + 2*q^191 + ...
%t A137677 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* _Michael Somos_, Oct 04 2015 *)
%o A137677 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n+1) - 1, x^(k^2 + 2*k) / prod(i=1, k, 1 - x^(4*i), 1 + x * O(x^(n - k^2 - 2*k)))), n))};
%Y A137677 Cf. A122130, A122134.
%K A137677 nonn
%O A137677 0,16
%A A137677 _Michael Somos_, Feb 04 2008