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

Author

N. J. A. Sloane

Keywords

• sign

Comments

The sequence shows the coefficients of sqrt(theta_3) regarded as an exponential generating function.

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

Cf. A000122 (theta_3), A015680, A186690.

Cf. A015665, A015666, A015667, A015669, A015671, A015672, A015673, A015675, A015676, A015677, A015678, A015679.

Programs

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 *)

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