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%I #22 Mar 28 2024 13:25:28
%S 1,-4,32,-832,41216,-3168256,359518208,-56319950848,11624409595904,
%T -3059387770077184,999955757611876352,-397353151288859164672,
%U 188655750511199441125376,-105472284295853235792510976,68582751548430569936978444288,-51320267059211655419226235076608
%N Expansion of the Jacobi elliptic function dn(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="/A370544/b370544.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^(n-k) for n >= 1, with a(0) = 1.
%F E.g.f. D(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) D(x) = dn(x,k) at k = 2.
%F (2.a) D(x) = cn(2*x, 1/2).
%F (2.b) D(x) = (4 - 8*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
%F (3) D(x) = 1 - Integral sqrt(1 - D(x)^2) * sqrt(3 + D(x)^2) dx.
%F (4) D(x) = cos( Integral sqrt(3 + D(x)^2) dx ).
%F (5.a) D(x) = sqrt(1 - 4*sn(x,2)^2).
%F (5.b) D(x) = sqrt(4*cn(x,2)^2 - 3).
%F O.g.f. 1/(1 + 4*x/(1 + 2^2*x/(1 + 4*3^2*x/(1 + 4^2*x/(1 + 4*5^2*x/(1 + 6^2*x/(1 + 4*7^2*x/(1 + ...)))))))) = 1 - 4*x + 32*x^2 - 832*x^3 + 41216*x^4 - 3168256*x^5 + ... (continued fraction, see Wall, 94.19, p. 374).
%F a(n) ~ (-1)^n * 2^(4*n+3) * 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.: D(x) = 1 - 4*x^2/2! + 32*x^4/4! - 832*x^6/6! + 41216*x^8/8! - 3168256*x^10/10! + 359518208*x^12/12! - 56319950848*x^14/14! + ...
%e where D(x) = dn(x,2).
%p # a(n) = (2*n)! * [x^(2*n)] dn(x, 2).
%p dn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiDN(z, k), z,
%p 2*len + 2), z, 2*n), n = 0..len) end:
%p dn_list(2, 15); # _Peter Luschny_, Mar 25 2024
%t nmax = 20;
%t DeleteCases[CoefficientList[JacobiDN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* _Jean-François Alcover_, Mar 28 2024 *)
%o (PARI) /* D(x) = Jacobi Elliptic Function dn(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(D,2*n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A028296 (dn(x,1)), A060627 (cn(x,k)).
%Y Cf. A370542 (sn(x,2)), A370543 (cn(x,2)), A249282.
%K sign
%O 0,2
%A _Paul D. Hanna_, Mar 25 2024