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%I #26 Mar 28 2024 13:41:14
%S 1,-1,17,-433,20321,-1584289,179967473,-28151779537,5812048858049,
%T -1529741412486721,499975227342256337,-198676311845589783793,
%U 94327947921149101192481,-52736138158762405338195169,34291374178966525773142501553,-25660133983889999165774819970577
%N Expansion of the Jacobi elliptic function cn(x,k) at k = 2 (even powers only).
%D H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.
%H Paul D. Hanna, <a href="/A370543/b370543.txt">Table of n, a(n) for n = 0..301</a>
%F a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^k for n >= 1, with a(0) = 1.
%F E.g.f. C(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
%F (1) C(x) = cn(x,k) at k = 2.
%F (2.a) C(x) = dn(2*x, 1/2).
%F (2.b) C(x) = (4 - 2*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
%F (3) C(x) = 1 - Integral sqrt(1 - C(x)^2) * sqrt(4*C(x)^2 - 3) dx.
%F (4) C(x) = cos( Integral sqrt(4*C(x)^2 - 3) dx ).
%F (5.a) C(x) = sqrt(1 - sn(x,2)^2).
%F (5.b) C(x) = sqrt(3 + dn(x,2)^2) / 2.
%F O.g.f. 1/(1 + x/(1 + 4*2^2*x/(1 + 3^2*x/(1 + 4*4^2*x/(1 + 5^2*x/(1 + 4*6^2*x/(1 + 7^2*x/(1 + ...)))))))) = 1 - x + 17*x^2 - 433*x^3 + 20321*x^4 - 1584289*x^5 + ... (continued fraction, see Wall, 94.18, p. 374). - [See formula in A060627 by Peter Bala, Apr 25 2017].
%F a(n) ~ (-1)^n * 2^(4*n+2) * agm(1,2)^(2*n+1) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - _Vaclav Kotesovec_, Mar 28 2024
%e E.g.f.: C(x) = 1 - x^2/2! + 17*x^4/4! - 433*x^6/6! + 20321*x^8/8! - 1584289*x^10/10! + 179967473*x^12/12! - 28151779537*x^14/14! + ...
%e where C(x) = cn(x,2).
%p # a(n) = (2*n)! * [x^(2*n)] cn(x, 2).
%p cn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiCN(z, k), z,
%p 2*len + 2), z, 2*n), n = 0..len) end:
%p cn_list(2, 15); # _Peter Luschny_, Mar 25 2024
%t nmax = 20;
%t DeleteCases[CoefficientList[JacobiCN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* _Jean-François Alcover_, Mar 28 2024 *)
%o (PARI) /* C(x) = Jacobi Elliptic Function cn(x,k) at k = 2: */
%o {a(n) = my(k=2,C=1,S=x,D=1); for(i=1,n,
%o S = intformal(C*D + x*O(x^(2*n+1)));
%o C = 1 - intformal(S*D);
%o D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(C,2*n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A028296 (cn(x,1)), A060627 (cn(x,k)).
%Y Cf. A370542 (sn(x,2)), A370544 (dn(x,2)), A249282.
%K sign
%O 0,3
%A _Paul D. Hanna_, Mar 25 2024
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