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 A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function. 1
 1, -72, 48384, -134120448, 1055796166656, -18987644270149632, 676784742282773397504, -43249455805185586718834688, 4599203617006025540525554139136, -768291761151281123722697889747566592, 192565676807771292904270021964021234663424 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is for the lemniscate case where g2=4, g3=0. - Michael Somos, Jul 10 2024 LINKS Table of n, a(n) for n=0..10. A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII. [Annotated scanned copy] See Eq. (16) and Table III. Tanay Wakhare and Christophe Vignat, Taylor coefficients of the Jacobi theta3(q) function, arXiv:1909.01508 [math.NT], 2019. FORMULA Hurwitz (Eq. (84)) gives a recurrence. a(n) = (-12)^n * A144849(n). - R. J. Mathar, Aug 03 2015 MAPLE A260779 := proc(n) option remember; if n = 0 then 1; else a :=0 ; for r from 0 to n-1 do s := n-1-r ; if s >=0 and s <= n-1 then a := a+procname(r)*procname(s) *binomial(4*n, 4*r+2) ; end if; end do: a*(-12) ; end if; end proc: # R. J. Mathar, Aug 03 2015 MATHEMATICA Block[{a}, a[n_] := If[n < 1, Boole[n == 0], Sum[Binomial[4 n, 4 j + 2] a[j] a[n - 1 - j], {j, 0, n - 1}]]; Array[(-12)^#*a[#] &, 11, 0]] (* Michael De Vlieger, Nov 20 2019, after Harvey P. Dale at A144849 *) a[ n_] := If[n<0, 0, With[{m = 4*n+2}, m!/2*SeriesCoefficient[ 1/WeierstrassP[u, {4, 0}], {u, 0, m}]]]; (* Michael Somos, Jul 10 2024 *) CROSSREFS Cf. A144849. Sequence in context: A292198 A177326 A276014 * A279656 A318184 A290182 Adjacent sequences: A260776 A260777 A260778 * A260780 A260781 A260782 KEYWORD sign,easy AUTHOR N. J. A. Sloane, Aug 02 2015 STATUS approved

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Last modified September 11 11:35 EDT 2024. Contains 375827 sequences. (Running on oeis4.)