A255252 Expansion of psi(x) * psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

1, -1, -1, 0, -2, 3, 2, 1, -1, -1, 1, -2, 1, -3, -2, -2, 3, 1, -1, 4, 3, -1, -1, 2, -4, 4, 1, 0, -1, -2, -3, -3, -4, 2, 3, -3, 0, 0, 5, 2, 0, -3, 2, -1, 4, 1, 0, 1, 3, 0, -2, 2, -1, -2, -4, -5, 2, 0, -7, 3, -4, 3, 1, 5, 2, -5, -1, -1, -3, 4, -1, 3, 4, 1, 4

Offset: 0

Author

Michael Somos, Feb 18 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x - x^2 - 2*x^4 + 3*x^5 + 2*x^6 + x^7 - x^8 - x^9 + x^10 + ...
G.f. = q^3 - q^11 - q^19 - 2*q^35 + 3*q^43 + 2*q^51 + q^59 - q^67 - q^75 + ...
		

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

Crossrefs

Cf. A034950.

Programs

Maple

A255252 := proc(n)
    local psi,x,i ;
    psi := add( A010054(i)*x^i,i=0..n) ;
    psi*subs(x=-x,psi)^2 ;
    coeftayl(%,x=0,n) ;
end proc:
seq(A255252(n),n=0..20) ; # R. J. Mathar, Feb 22 2021

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / (4 x^(3/8)), {x, 0, n}];

PARI

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

Formula

Expansion of f(-x) * f(-x^4)^2 in powers of x where f() is a Ramanujan theta function.

Expansion of q^(-3/8) * eta(q) * eta(q^4)^2 in powers of q.

Euler transform of period 4 sequence [ -1, -1, -1, -3, ...].

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2.

2 * a(n) = A034950(4*n + 1).