A213267 -id:A213267 - cp's OEIS frontend

A132302 Expansion of f(-x, -x^5) * f(-x^6) / f(-x)^2 in powers of x where f(, ) and f() are Ramanujan theta functions.

1, 1, 3, 5, 10, 15, 26, 39, 63, 92, 140, 201, 295, 415, 591, 818, 1140, 1554, 2126, 2861, 3855, 5126, 6816, 8970, 11793, 15372, 20007, 25857, 33356, 42771, 54734, 69683, 88530, 111968, 141312, 177642, 222842, 278557, 347484, 432095, 536230, 663549, 819504

Offset: 0

Michael Somos, Aug 17 2007

nonn

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

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 26*x^6 + 39*x^7 + ...
G.f. = q + q^3 + 3*q^5 + 5*q^7 + 10*q^9 + 15*q^11 + 26*q^13 + 39*q^15 + ...

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

Cf. A124243, A132180, A132301, A132975, A213267.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6]^2 / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))};

Expansion of q^(-1/2) * eta(q^6)^3 / (eta(q) * eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^2) * q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * v^4 * (v - 3*u^2) * (2*v - 3*u^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132301.
a(n) = A124243(2*n + 1) = A132180(2*n + 1) = A132975(2*n + 1) = A213267(2*n + 1). - Michael Somos, Nov 01 2015
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4)*3^(3/2)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A132975 Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548

Offset: 1

Michael Somos, Sep 07 2007

nonn

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

			G.f. = q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10001
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S115.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A128129, A128640, A132302, A132972, A132976. Essentially the same as A213267.

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(9*k)) * (1+x^(18*k)) / (1-x^(4*k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)), n))};

Expansion of eta(q^2) * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * u2 - (1 + u1 + u2) * (u3 + u6 + 3 * u3 * u6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132976.
G.f.: x * Product_{k>0} P(3,x^k) * P(9,x^k) * P(12,x^k) * P(36,x^k) where P(n,x) is the n-th cyclotomic polynomial.
3 * a(n) = A132972(n) unless n=0. a(2*n) = A128129(n). a(2*n + 1) = A132302(n). a(3*n) = A128640(n). Convolution inverse of A132976.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

cp's OEIS Frontend

A132302 Expansion of f(-x, -x^5) * f(-x^6) / f(-x)^2 in powers of x where f(, ) and f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula