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

0,5

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 = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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).

Example

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 + ...

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}];

Prog

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

Crossrefs

Cf. A034950.

Sequence in context: A211994 A122402 A179008 * A174985 A008406 A039735

Adjacent sequences: A255249 A255250 A255251 * A255253 A255254 A255255

Keyword

sign

Author

Michael Somos, Feb 18 2015

Status

approved