cp's OEIS Frontend

A255365 Expansion of phi(-x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A255365 #11 Jun 27 2025 23:06:57
%S A255365 1,0,0,-2,0,0,0,0,0,0,0,0,2,0,0,0,1,0,0,-2,0,0,0,0,0,0,0,-2,2,0,0,0,1,
%T A255365 0,0,-2,0,0,0,0,0,0,0,-2,2,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,2,0,
%U A255365 0,0,0,0,0,0,0,0,0,-2,0,0,0,0,3,0,0,-2
%N A255365 Expansion of phi(-x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A255365 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A255365 G. C. Greubel, <a href="/A255365/b255365.txt">Table of n, a(n) for n = 0..1000</a>
%H A255365 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A255365 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A255365 Expansion of q^(-2/3) * eta(q^3)^2 * eta(q^32) * eta(q^48)^2 / (eta(q^6) * eta(q^16) * eta(q^96)) in powers of q.
%F A255365 Euler transform of a period 96 sequence.
%F A255365 a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.
%F A255365 a(3*n) = A002448(n). a(3*n + 32) = A227395(n).
%e A255365 G.f. = 1 - 2*x^3 + 2*x^12 + x^16 - 2*x^19 - 2*x^27 + 2*x^28 + x^32 + ...
%e A255365 G.f. = q^2 - 2*q^11 + 2*q^38 + q^50 - 2*q^59 - 2*q^83 + 2*q^86 + q^98 + ...
%t A255365 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^16, x^16] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^48], {x, 0, n}];
%t A255365 a[n_]:=SeriesCoefficient[EllipticTheta[3,0,-x^3]*EllipticTheta[3,0,-x^48 ]/QPochhammer[x^16, x^32], {x, 0, n}]; Table[a[n], {n,0,100}] (* _G. C. Greubel_, Mar 14 2018 *)
%o A255365 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^6 + A) * eta(x^16 + A) * eta(x^96 + A)), n))};
%o A255365 (PARI) {a(n) = my(A, p, e); if( n<0 || n%4==1, 0, n = 3*n+2; A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -(e==1), if( p==3, 0, if( p%8 < 4, e+1, 1-e%2))))))};
%Y A255365 Cf. A002448, A227395.
%K A255365 sign
%O A255365 0,4
%A A255365 _Michael Somos_, Feb 21 2015