OFFSET
0,4
COMMENTS
The sequence shows the coefficients of sqrt(theta_3) regarded as an exponential generating function.
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..449
FORMULA
E.g.f. appears to equal exp( Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 + x^(2*n+1))) ). - Peter Bala, Dec 23 2021
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A186690(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Jul 07 2023
EXAMPLE
sqrt(theta_3) = 1 + q - (1/2)*q^2 + (1/2)*q^3 + (3/8)*q^4 - (1/8)*q^5 + (3/16)*q^6 - (7/16)*q^7 + (67/128)*q^8 + (27/128)*q^9 + ...
MAPLE
# get basic theta series in maple
maxd:=201:
# get th2, th3, th4 = Jacobi theta constants out to degree maxd
temp0:=trunc(evalf(sqrt(maxd)))+2:
a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:
th2:=series(a, q, maxd); # A098108
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:
th3:=series(a, q, maxd); # A000122
th4:=series(subs(q=-q, th3), q, maxd); # A002448
series(sqrt(th3), q, maxd); # this sequence
MATHEMATICA
nmax = 25; CoefficientList[Series[EllipticTheta[3, 0, x]^(1/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 23 2018 *)
CROSSREFS
KEYWORD
sign
AUTHOR
EXTENSIONS
Entry revised by N. J. A. Sloane, Oct 22 2018
STATUS
approved