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 A096920 Expansion of q^(-1/12) * eta(q^2)^4 / (eta(q)^2 * eta(q^4)) in powers of q. 2
 1, 2, 1, 2, 3, 2, 4, 4, 4, 6, 7, 8, 8, 10, 11, 14, 16, 16, 20, 22, 24, 28, 32, 34, 39, 44, 48, 54, 60, 66, 73, 82, 88, 98, 108, 118, 132, 144, 156, 172, 188, 204, 224, 244, 265, 290, 316, 340, 372, 404, 436, 474, 513, 554, 600, 650, 700, 756, 816, 878, 948, 1022, 1096, 1182 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000700(). E.g., a(8) = b(8)+b(7)+b(5)+b(2) = 2+1+1+0 = 4. G.f.: Product_{k>0} (1 - x^(4*k)) * (1 + x^(2*k-1))^2. - Michael Somos, Mar 25 2008 Expansion of psi(q) * chi(q) = f(q) / chi(-q) = f(q)^2 / phi(-q^2) = phi(-q^2) / chi(-q)^2 = phi(q) / chi(-q^2) = psi(q)^2 / f(-q^4) = f(-q^4) * chi(q)^2 where phi(), psi(), chi(), f() are Ramanujan theta functions. Euler transform of period 4 sequence [ 2, -2, 2, -1, ...]. - Michael Somos, Mar 25 2008 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 12^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138559. a(n) ~ exp(Pi*sqrt(n/6)) / (2*sqrt(2*n)). - Vaclav Kotesovec, Sep 07 2015 EXAMPLE q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 4*q^73 + 4*q^85 + 4*q^97 + ... MATHEMATICA nmax = 40; CoefficientList[Series[Product[(1 - x^(4*k)) * (1 + x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *) eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/12) *eta[q^2]^4/(eta[q]^2*eta[q^4]), {q, 0, 50}], q]; Table[a[[n]], {n, 0, 50}] (* G. C. Greubel, May 09 2018 *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / eta(x + A)^2 / eta(x^4 + A), n))} /* Michael Somos, Mar 25 2008 */ CROSSREFS Sequence in context: A080845 A290370 A029166 * A087154 A029839 A082304 Adjacent sequences: A096917 A096918 A096919 * A096921 A096922 A096923 KEYWORD easy,nonn AUTHOR Vladeta Jovovic, Aug 18 2004 STATUS approved

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Last modified March 25 12:08 EDT 2023. Contains 361521 sequences. (Running on oeis4.)