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%I #23 Jul 09 2022 11:07:56
%S 1,11069,4494351,834687179,109645021894,11966116940238,
%T 1171517154238290,107266611330420090,9412382749388124015,
%U 803475280086029066515,67362921649153881472361,5581153512072331417781229
%N Coefficients of Jacobi elliptic function c(4,m).
%H A. Fransen, <a href="http://dx.doi.org/10.1090/S0025-5718-1981-0628708-X">Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k)</a>, Math. Comp., 37 (1981), 475-497.
%H S. Wrigge, <a href="https://doi.org/10.1090/S0025-5718-1981-0606513-8">Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function sn(x, k)</a>, Math. Comp. 36 (1981), 555-564. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010]
%F a(n) = (3^(4*n+19) - (24*n+36)*7^(2*n+9) + (96*n^2+312*n+150)*5^(2*n+9) - (256*n^3+1344*n^2+1520*n+117)*3^(2*n+9) + 512*n^4+3584*n^3+6208*n^2+1336*n-846) / 196608. - _Vaclav Kotesovec_ after Fransen, Jul 30 2013
%p a:=proc(n) options remember: local m: if n>2 then if n mod 2 = 0 then m:=n/2-1: RETURN(-4*(1+k^2)*a(n-2)+6*k^2*add(binomial(n-2,2*v)*a(2*v)*a(n-2-2*v),v=1..m-1)) else m:=(n-1)/2-1: RETURN(-(1+k^2)*a(n-2)+2*k^2*add(binomial(n-2,2*v+1)*a(2*v+1)*a(n-3-2*v),v=0..m-1)) fi else RETURN([1,2][n]) fi:end: seq(abs(coeff(a(2*i+1),k,8)),i=4..23); # Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
%t a[n_] := a[n] = Module[{m}, If[n > 2, If[Mod[n, 2] == 0, m = n/2 - 1; Return[-4*(1 + k^2)*a[n - 2] + 6*k^2*Sum[Binomial[n - 2, 2*v]*a[2*v]*a[n - 2 - 2*v], {v, 1, m - 1}]], m = (n - 1)/2 - 1; Return[-(1 + k^2)*a[n - 2] + 2*k^2*Sum[Binomial[n - 2, 2*v + 1]*a[2*v + 1]*a[n - 3 - 2*v], {v, 0, m - 1}]]], Return[{1, 2}[[n]]]]];
%t Table[Abs[Coefficient[a[2*i + 1], k, 8]], {i, 4, 15}] (* _Jean-François Alcover_, Jul 08 2022, after Herman Jamke's Maple code *)
%Y Cf. A060928 (4th lower diagonal).
%K nonn
%O 0,2
%A _Simon Plouffe_
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
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