A006195 Reversion of Jacobi theta_3.

1, -2, 8, -40, 222, -1316, 8160, -52272, 343220, -2297682, 15623760, -107611608, 749209832, -5264005060, 37277153920, -265788870480, 1906489923022, -13747860118724, 99606357848920, -724732875917064, 5293303253527704, -38795196044205056

Offset

0,2

Links

Table of n, a(n) for n=0..21.

J.-G. Penaud, Arbres et Animaux, Ph.D. Dissertation, Université Bordeaux I, 1990, cover page and p. 76. (Annotated scanned copy)

Index entries for reversions of series

Formula

REVERT(A000122).

From Vaclav Kotesovec, Sep 27 2023: (Start)

G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + (-x)^k*A(x)^k)/(1 - (-x)^k*A(x)^k).

a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 7.86298339570590526151934790995382716105758424871057843176888470144337... and c = 0.617020565581840591336246430220953133238702598666548444780767269...

(End)

Maple

# Using function CompInv from A357588.
CompInv(22, n -> if n = 1 then 1 elif issqr(n-1) then 2 else 0 fi); # Peter Luschny, Oct 05 2022

Mathematica

nmax = 30; A[_] = 0; Do[A[x_] = Product[(1 + x^k*A[x]^k)/(1 - x^k*A[x]^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * (-1)^Range[0, nmax] (* Vaclav Kotesovec, Sep 27 2023 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[-1, r*s] == 2*s*QPochhammer[r*s], (2* QPochhammer[r*s]*(-Log[r*s] + Log[1 - r*s] + QPolyGamma[0, 1, r*s])) / Log[r*s] + r*(Derivative[0, 1][QPochhammer][-1, r*s] - 2*s*Derivative[0, 1][QPochhammer][r*s, r*s]) == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 27 2023 *)

Prog

(PARI) N=66; x='x+O('x^N); /* that many terms */
Vec(serreverse(x*sum(n=-N, N, x^(n^2)))) /* show terms */ /* Joerg Arndt, May 25 2011 */

Crossrefs

Cf. A000122, A109085, A181315.

Sequence in context: A234938 A143388 A027282 * A214763 A219587 A092807

Adjacent sequences: A006192 A006193 A006194 * A006196 A006197 A006198

Keyword

sign

Author

N. J. A. Sloane

Extensions

Signs corrected by N. J. A. Sloane, Dec 24 2001

Status

approved