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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 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 Cf. A000122 (theta_3), A015680, A186690. Cf. A015665, A015666, A015667, A015669, A015671, A015672, A015673, A015675, A015676, A015677, A015678, A015679. Sequence in context: A242438 A355716 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 1 14:44 EDT 2023. Contains 365826 sequences. (Running on oeis4.)