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A187615 Expansion of f(-x^17, -x^19) + x^4 * f(-x, -x^35) in powers of x where f(,) is Ramanujan's general theta function.

%I A187615 #15 Feb 16 2025 08:33:14
%S A187615 1,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A187615 0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A187615 0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A187615 Expansion of f(-x^17, -x^19) + x^4 * f(-x, -x^35) in powers of x where f(,) is Ramanujan's general theta function.
%C A187615 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A187615 G. C. Greubel, <a href="/A187615/b187615.txt">Table of n, a(n) for n = 0..1000</a>
%H A187615 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A187615 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A187615 Euler transform of period 36 sequence [ 0, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, -1, ...].
%F A187615 G.f.: Sum_{k in Z} (-1)^k * x^(18*k^2) * (x^(17*k + 4) + x^k).
%F A187615 a(n) = A133985(3*n). a(5*n + 1) = a(5*n + 3) = 0.
%e A187615 G.f. = 1 + x^4 - x^5 - x^17 - x^19 - x^39 + x^42 + x^70 + x^74 + x^110 + ...
%e A187615 G.f. = q + q^289 - q^361 - q^1225 - q^1369 - q^2809 + q^3025 + q^5041 + q^5329 + ...
%t A187615 QP:= QPochhammer; a[n_]:= SeriesCoefficient[QP[q^36, q^36]*(QP[q^17, q^36]*QP[q^19, q^36] + q^4*QP[q, q^36]*QP[q^35, q^36]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Apr 04 2018 *)
%o A187615 (PARI) {a(n) = my(m); if( n<0, 0, if( issquare( 72*n + 1, &m), if( m%36 != 1, m=-m); if( m%36 == 1, m = (m-1) / 36, if( m%36 != 19, m=-m); if( m%36! = 19, return(0));  m = (m+17) / 36); (-1)^m))};
%o A187615 (PARI) {a(n) = my(A); if( n<0, 0, n*=3; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^5 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^2, n))};
%Y A187615 Cf. A133985.
%K A187615 sign
%O A187615 0,1
%A A187615 _Michael Somos_, Mar 11 2011