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 A138517 Expansion of (phi(-q^5) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function. 4
 1, 4, 12, 32, 76, 164, 336, 656, 1228, 2228, 3932, 6768, 11408, 18872, 30688, 49152, 77644, 121096, 186684, 284720, 429916, 643168, 953904, 1403312, 2048784, 2969764, 4275656, 6116480, 8696864, 12294680, 17285776, 24176288, 33645132 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) := Product_{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) := Product_{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 Expansion of ( (eta(q^5) / eta(q))^2 * eta(q^2) / eta(q^10) )^2 in powers of q. Euler transform of period 10 sequence [ 4, 2, 4, 2, 0, 2, 4, 2, 4, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u - 1) - 4 * u * v * (v - 1). G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (1 - 5*u) * v * (1 - v) * (1 - 5*v). G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138516. G.f.: (Product_{k>0} P(5, x^k) / P(10, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial. a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018 Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 3/10 + (1/10)*sqrt(5) + (1/10)*sqrt(10 + 6*sqrt(5)). - Simon Plouffe, Mar 04 2021 EXAMPLE 1 + 4*q + 12*q^2 + 32*q^3 + 76*q^4 + 164*q^5 + 336*q^6 + 656*q^7 + ... MATHEMATICA eta[x_] := x^(1/24)*QPochhammer[x]; A138517[n_] := SeriesCoefficient[ ((eta[q^5]/eta[q])^2*eta[q^2]/eta[q^10])^2, {q, 0, n}]; Table[ A138517[n], {n, 0, 50}] (* G. C. Greubel, Sep 29 2017 *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^5 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^10 + A))^2, n))} CROSSREFS Cf. 4 * A095846(n) = a(n) unless n=0. Convolution inverse of A138518. Convolution square of A138526. Sequence in context: A233447 A127811 A361099 * A001934 A004403 A084566 Adjacent sequences: A138514 A138515 A138516 * A138518 A138519 A138520 KEYWORD nonn AUTHOR Michael Somos, Mar 23 2008 STATUS approved

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Last modified September 15 20:25 EDT 2024. Contains 375955 sequences. (Running on oeis4.)