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 A210458 Expansion of q * (psi(-q^5) / psi(-q))^2 in powers of q where psi() is a Ramanujan theta function. 5
 1, 2, 3, 6, 11, 16, 24, 38, 57, 82, 117, 168, 238, 328, 448, 614, 834, 1114, 1480, 1966, 2592, 3384, 4398, 5704, 7361, 9436, 12045, 15344, 19470, 24576, 30922, 38822, 48576, 60548, 75259, 93342, 115454, 142360, 175104, 214958, 263262, 321584, 391993, 476952 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (eta(q^2) * eta(q^5) * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10)))^2 in powers of q. Euler transform of period 20 sequence [ 2, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, ...]. 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 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. of A145740. G.f.: x * (Product_{k>0} P(5, x^k) * P(20, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial. A138519(n) = -(-1)^n * a(n). Convolution inverse of A145740. a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015 Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = -2/5 + (1/5)*sqrt(5). - Simon Plouffe, Mar 02 2021 EXAMPLE q + 2*q^2 + 3*q^3 + 6*q^4 + 11*q^5 + 16*q^6 + 24*q^7 + 38*q^8 + 57*q^9 + ... MATHEMATICA nmax=60; CoefficientList[Series[Product[((1+x^k) * (1-x^(5*k)) * (1+x^(10*k)) / (1-x^(4*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *) a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, I*q^(5/2)]/EllipticTheta[ 2, 0, I*Sqrt[q]])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Dec 07 2017 *) PROG (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)))^2, n))} CROSSREFS Cf. A138519, A145740. Sequence in context: A351203 A138519 A138520 * A228864 A289434 A049794 Adjacent sequences: A210455 A210456 A210457 * A210459 A210460 A210461 KEYWORD nonn AUTHOR Michael Somos, Jan 21 2013 STATUS approved

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Last modified July 22 22:03 EDT 2024. Contains 374544 sequences. (Running on oeis4.)