OFFSET
0,5
COMMENTS
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
KEYWORD
sign
AUTHOR
Michael Somos, Feb 18 2015
STATUS
approved