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 A138519 Expansion of q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function. 6
 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 (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 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of ((eta(q^10) / eta(q^2))^2 * eta(q) / eta(q^5))^2 in powers of q. Euler transform of period 10 sequence [ -2, 2, -2, 2, 0, 2, -2, 2, -2, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - v * (1 - u) * (1 - 5*u). 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 the g.f. for A138518. G.f.: x * (Product_{k>0} P(5, x^k) * P(10, x^k)^2)^2 where P(n, x) is the n-th cyclotomic polynomial. a(n) = - A138520(n) unless n=0. -5 * a(n) = A138521(n) unless n=0. Convolution inverse of A138516. a(n) = -(-1)^n * A210458(n). - Michael Somos, Sep 16 2015 a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017 EXAMPLE G.f. = 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 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(5/2)] / EllipticTheta[ 2, 0, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Sep 16 2015 *) PROG (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x + A) / eta(x^5 + A) * ( eta(x^10 + A) / eta(x^2 + A) )^2)^2, n))}; CROSSREFS Cf. A138516, A138520, A138521, A210458. Sequence in context: A298702 A076307 A102990 * A138520 A210458 A228864 Adjacent sequences:  A138516 A138517 A138518 * A138520 A138521 A138522 KEYWORD sign AUTHOR Michael Somos, Mar 23 2008 STATUS approved

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Last modified February 25 10:41 EST 2020. Contains 332228 sequences. (Running on oeis4.)