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A015664 Expansion of e.g.f. theta_3^(1/2). 2
1, 1, -1, 3, 9, -15, 135, -2205, 21105, 76545, 694575, -6392925, -56600775, 66891825, -19964169225, 741313447875, 5375639894625, 44667168170625, -2328500019470625, 5663134786183875, -466442955127524375, 11513119609487120625 (list; graph; refs; listen; history; text; internal format)
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

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

Cf. A000122 (theta_3), A015680.

Sequence in context: A348199 A242438 A083556 * A272621 A134137 A174179

Adjacent sequences:  A015661 A015662 A015663 * A015665 A015666 A015667

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry revised by N. J. A. Sloane, Oct 22 2018

STATUS

approved

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)