A138521 -id:A138521 - cp's OEIS frontend

A138519 Expansion of q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.

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

Offset: 1

Michael Somos, Mar 23 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)

			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 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A138516, A138520, A138521, A210458.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(5/2)] / EllipticTheta[ 2, 0, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Sep 16 2015 *)

{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))};

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

A225701 Expansion of chi(q)^5 / chi(q^5) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 5, 10, 15, 30, 55, 80, 120, 190, 285, 410, 585, 840, 1190, 1640, 2240, 3070, 4170, 5570, 7400, 9830, 12960, 16920, 21990, 28520, 36805, 47180, 60225, 76720, 97350, 122880, 154610, 194110, 242880, 302740, 376295, 466710, 577270, 711800, 875520, 1074790

Offset: 0

Michael Somos, May 17 2013

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 5*q + 10*q^2 + 15*q^3 + 30*q^4 + 55*q^5 + 80*q^6 + 120*q^7 + 190*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..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

Cf. A138521, A210458, A223903.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^5 / QPochhammer[ -q^5, q^10], {q, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^k)^5 * (1+x^k)^10 * (1+x^(10*k)) / ((1-x^(4*k))^5 * (1+x^(5*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A)^5 * eta(x^4 + A)^5 * eta(x^10 + A)^2), n))};

Expansion of eta(q^2)^10 * eta(q^5) * eta(q^20) / (eta(q)^5 * eta(q^4)^5 * eta(q^10)^2) in powers of q.
Euler transform of period 20 sequence [ 5, -5, 5, 0, 4, -5, 5, 0, 5, -4, 5, 0, 5, -5, 4, 0, 5, -5, 5, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A223903.
a(n) = (-1)^n * A138521(n). a(n) = 5 * A210458(n) unless n=0.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
Empirical: Sum_{n>=0} a(n)/exp(Pi*n) = sqrt(5) - 1. - Simon Plouffe, Mar 02 2021

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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.

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