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A015664
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Expansion of e.g.f. theta_3^(1/2).
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2
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1, 1, -1, 3, 9, -15, 135, -2205, 21105, 76545, 694575, -6392925, -56600775, 66891825, -19964169225, 741313447875, 5375639894625, 44667168170625, -2328500019470625, 5663134786183875, -466442955127524375, 11513119609487120625
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OFFSET
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0,4
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COMMENTS
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The sequence shows the coefficients of sqrt(theta_3) regarded as an exponential generating function.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..449
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FORMULA
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O.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
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EXAMPLE
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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 + ...
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MAPLE
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# 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
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nmax = 25; CoefficientList[Series[EllipticTheta[3, 0, x]^(1/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 23 2018 *)
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CROSSREFS
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Cf. A000122 (theta_3), A015680.
Sequence in context: A242438 A355716 A083556 * A272621 A134137 A174179
Adjacent sequences: A015661 A015662 A015663 * A015665 A015666 A015667
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Oct 22 2018
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STATUS
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approved
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