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A122190 Expansion of q^(-1/4) * eta(q^2) * eta(q^5)^3 / (eta(q) * eta(q^10)) in powers of q.

%I A122190 #17 Feb 16 2025 08:33:02
%S A122190 1,1,1,2,2,0,1,2,0,2,2,1,1,2,0,2,2,0,2,0,1,2,2,0,2,2,0,2,2,2,1,1,0,0,
%T A122190 2,0,2,2,2,2,0,0,3,2,0,2,2,0,2,2,0,2,0,0,0,4,1,2,2,0,2,1,0,0,2,2,2,2,
%U A122190 0,2,2,0,3,2,0,0,2,0,2,2,0,2,0,2,2,0,0,2,2,0,1,2,2,2,4,0,0,2,0,2,2,1,2,0,0
%N A122190 Expansion of q^(-1/4) * eta(q^2) * eta(q^5)^3 / (eta(q) * eta(q^10)) in powers of q.
%C A122190 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A122190 Convolution product of A133100 and A133101. - _Michael Somos_, Feb 10 2015
%H A122190 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A122190 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A122190 Euler transform of period 10 sequence [ 1, 0, 1, 0, -2, 0, 1, 0, 1, -2, ...].
%F A122190 a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(5^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
%F A122190 G.f.: Product_{k>0} (1 - x^(5*k))^2 * (1 + x^k) / (1 + x^(5*k)).
%F A122190 G.f.: Sum_{k>=0} a(k) * x^(4k+1) = Sum_{k>0 odd} x^k * (1 - x^(2*k)) * (1 - x^(6*k)) / (1 + x^(10*k)).
%F A122190 Expansion of f(x, x^4) * f(x^2, x^3) in powers of x where f() is the Ramanujan two-variable theta function.
%F A122190 Expansion of psi(x)^2 - x * psi(x^5)^2 in powers of x where psi() is a Ramanujan theta function.
%F A122190 G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133573.
%F A122190 a(n) = A053694(4*n) = A094247(4*n + 1).
%F A122190 a(3*n + 2) = a(5*n + 1) = a(n). - _Michael Somos_, Feb 10 2015
%F A122190 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/5 = 1.256637... (A019694). - _Amiram Eldar_, Dec 29 2023
%e A122190 G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + x^11 + ...
%e A122190 G.f. = q + q^5 + q^9 + 2*q^13 + 2*q^17 + q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...
%t A122190 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^5]^3 / (QPochhammer[ x] QPochhammer[ x^10]), {x, 0, n}]; (* _Michael Somos_, Feb 10 2015 *)
%o A122190 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^5 + A)^3 / (eta(x + A) * eta(x^10 + A)), n))};
%o A122190 (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 4*n + 1; A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==5, 1, if( p%4==1, e+1, (1 + (-1)^e) / 2))))))};
%Y A122190 Cf. A019694, A053694, A094247, A133100, A133101, A133573.
%Y A122190 Cf. A000122, A000700, A010054, A121373.
%K A122190 nonn,easy
%O A122190 0,4
%A A122190 _Michael Somos_, Aug 24 2006