A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.

1, -72, 48384, -134120448, 1055796166656, -18987644270149632, 676784742282773397504, -43249455805185586718834688, 4599203617006025540525554139136, -768291761151281123722697889747566592, 192565676807771292904270021964021234663424

Offset: 0

Author

N. J. A. Sloane, Aug 02 2015

Keywords

• sign
• easy

Comments

This is for the lemniscate case where g2=4, g3=0. - Michael Somos, Jul 10 2024

Links

• 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.

Crossrefs

Cf. A144849.

Programs

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 *)

Formula

Hurwitz (Eq. (84)) gives a recurrence.

a(n) = (-12)^n * A144849(n). - R. J. Mathar, Aug 03 2015