cp's OEIS Frontend

A260089 Expansion of psi(x^2) * f(x, x^2) in powers of x where psi(), f() are Ramanujan theta functions.

%I A260089 #9 Feb 16 2025 08:33:26
%S A260089 1,1,2,1,1,1,1,3,1,1,0,1,2,2,2,1,0,2,1,1,1,2,2,0,2,1,1,2,2,0,1,1,3,0,
%T A260089 1,3,0,2,1,0,1,1,4,1,1,1,2,2,0,1,0,1,2,1,1,1,2,4,1,1,1,1,0,3,1,1,0,0,
%U A260089 2,1,2,2,2,1,1,0,1,4,1,2,0,1,3,1,1,0,0
%N A260089 Expansion of psi(x^2) * f(x, x^2) in powers of x where psi(), f() are Ramanujan theta functions.
%C A260089 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260089 G. C. Greubel, <a href="/A260089/b260089.txt">Table of n, a(n) for n = 0..1000</a>
%H A260089 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260089 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260089 Expansion of phi(-x^3) * f(-x^4)^2 / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
%F A260089 Expansion of q^(-7/24) * eta(q^3)^2 * eta(q^4)^2 / (eta(q) * eta(q^6)) in powers of q.
%F A260089 Euler transform of period 12 sequence [ 1, 1, -1, -1, 1, 0, 1, -1, -1, 1, 1, -2, ...].
%F A260089 G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) / (1 - x^k + x^(2*k)).
%e A260089 G.f. = 1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6 + 3*x^7 + x^8 + x^9 + ...
%e A260089 G.f. = q^7 + q^31 + 2*q^55 + q^79 + q^103 + q^127 + q^151 + 3*q^175 + ...
%t A260089 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ x^4]^2 / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}];
%o A260089 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};
%K A260089 nonn
%O A260089 0,3
%A A260089 _Michael Somos_, Jul 15 2015