A015664 Expansion of e.g.f. theta_3^(1/2).
1, 1, -1, 3, 9, -15, 135, -2205, 21105, 76545, 694575, -6392925, -56600775, 66891825, -19964169225, 741313447875, 5375639894625, 44667168170625, -2328500019470625, 5663134786183875, -466442955127524375, 11513119609487120625
Offset: 0
Keywords
Examples
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 + ...
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
Crossrefs
Programs
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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
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Mathematica
nmax = 25; CoefficientList[Series[EllipticTheta[3, 0, x]^(1/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 23 2018 *)
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
Extensions
Entry revised by N. J. A. Sloane, Oct 22 2018
Comments