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A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A113419 #31 Feb 16 2025 08:32:59
%S A113419 1,2,-4,-8,7,10,-12,-8,18,18,-16,-24,21,20,-28,-32,20,32,-36,-24,42,
%T A113419 42,-28,-48,57,36,-52,-40,36,58,-60,-56,48,66,-48,-72,74,42,-80,-80,
%U A113419 61,82,-72,-56,90,96,-64,-72,98,70,-100,-104,64,106,-108,-72,114,96
%N A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A113419 Number 46 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A113419 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A113419 G. C. Greubel, <a href="/A113419/b113419.txt">Table of n, a(n) for n = 0..1000</a>
%H A113419 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A113419 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A113419 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A113419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A113419 Expansion of q^(-1/2) * (eta(q^2)^9 * eta(q^8)^2) / (eta(q)^2 * eta(q^4)^5) in powers of q. - _Michael Somos_, Mar 14 2012
%F A113419 Euler transform of period 8 sequence [ 2, -7, 2, -2, 2, -7, 2, -4, ...].
%F A113419 a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * (-1)^floor(p/4) and y = (-1)^floor(p/2).
%F A113419 G.f.: Sum_{k>0} (2*k - 1) * (-1)^[(k - 1)/2] * x^(2*k - 1) / (1 + x^(4*k - 2)).
%F A113419 a(n) = (-1)^n * A113417(n) = (-1)^floor(n/2) * A258096(n) = (-1)^(n + floor(n/2)) * A209940(n).
%F A113419 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109039. - _Michael Somos_, May 20 2015
%e A113419 G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 7*x^4 + 10*x^5 - 12*x^6 - 8*x^7 + 18*x^8 + ...
%e A113419 G.f. = q + 2*q^3 - 4*q^5 - 8*q^7 + 7*q^9 + 10*q^11 - 12*q^13 - 8*q^15 + ...
%t A113419 a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^9 QPochhammer[ q^8]^2 / (QPochhammer[ q]^2 QPochhammer[ q^4]^5), {q, 0, n}]; (* _Michael Somos_, May 19 2015 *)
%t A113419 a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* _Michael Somos_, May 19 2015 *)
%o A113419 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, d*(d%2) * (-1)^((n/d)\2 + (d-1)\4)))};
%o A113419 (PARI) {a(n) = my(A, p, e, t); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, t = (-1)^(p\2); p *= kronecker( -2, p); (p^(e+1) - t^(e+1)) / (p - t) )))};
%o A113419 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^5), n))};
%Y A113419 Cf. A109039, A113417, A209940, A258096.
%K A113419 sign
%O A113419 0,2
%A A113419 _Michael Somos_, Oct 29 2005