A370542 -id:A370542 - cp's OEIS frontend

A370543 Expansion of the Jacobi elliptic function cn(x,k) at k = 2 (even powers only).

1, -1, 17, -433, 20321, -1584289, 179967473, -28151779537, 5812048858049, -1529741412486721, 499975227342256337, -198676311845589783793, 94327947921149101192481, -52736138158762405338195169, 34291374178966525773142501553, -25660133983889999165774819970577

Offset: 0

Paul D. Hanna, Mar 25 2024

sign

			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! + ...
where C(x) = cn(x,2).

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A028296 (cn(x,1)), A060627 (cn(x,k)).

Cf. A370542 (sn(x,2)), A370544 (dn(x,2)), A249282.

# a(n) = (2*n)! * [x^(2*n)] cn(x, 2).
cn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiCN(z, k), z,
2*len + 2), z, 2*n), n = 0..len) end:
cn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiCN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* Jean-François Alcover, Mar 28 2024 *)

/* C(x) = Jacobi Elliptic Function cn(x,k) at k = 2: */
{a(n) = my(k=2,C=1,S=x,D=1); for(i=1,n,
S = intformal(C*D + x*O(x^(2*n+1)));
C = 1 - intformal(S*D);
D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(C,2*n)}
for(n=0,20,print1(a(n),", "))

a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^k for n >= 1, with a(0) = 1.
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.
(1) C(x) = cn(x,k) at k = 2.
(2.a) C(x) = dn(2*x, 1/2).
(2.b) C(x) = (4 - 2*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
(3) C(x) = 1 - Integral sqrt(1 - C(x)^2) * sqrt(4*C(x)^2 - 3) dx.
(4) C(x) = cos( Integral sqrt(4*C(x)^2 - 3) dx ).
(5.a) C(x) = sqrt(1 - sn(x,2)^2).
(5.b) C(x) = sqrt(3 + dn(x,2)^2) / 2.
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].
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

A370544 Expansion of the Jacobi elliptic function dn(x,k) at k = 2 (even powers only).

Original entry on oeis.org

1, -4, 32, -832, 41216, -3168256, 359518208, -56319950848, 11624409595904, -3059387770077184, 999955757611876352, -397353151288859164672, 188655750511199441125376, -105472284295853235792510976, 68582751548430569936978444288, -51320267059211655419226235076608

Offset: 0

Paul D. Hanna, Mar 25 2024

sign

			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! + ...
where D(x) = dn(x,2).

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A028296 (dn(x,1)), A060627 (cn(x,k)).

Cf. A370542 (sn(x,2)), A370543 (cn(x,2)), A249282.

# a(n) = (2*n)! * [x^(2*n)] dn(x, 2).
dn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiDN(z, k), z,
2*len + 2), z, 2*n), n = 0..len) end:
dn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiDN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* Jean-François Alcover, Mar 28 2024 *)

/* D(x) = Jacobi Elliptic Function dn(x,k) at k = 2: */
{a(n) = my(k=2, C=1,S=x,D=1); for(i=1,n,
S = intformal(C*D + x*O(x^(2*n+1)));
C = 1 - intformal(S*D);
D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(D,2*n)}
for(n=0,20,print1(a(n),", "))

a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^(n-k) for n >= 1, with a(0) = 1.
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.
(1) D(x) = dn(x,k) at k = 2.
(2.a) D(x) = cn(2*x, 1/2).
(2.b) D(x) = (4 - 8*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
(3) D(x) = 1 - Integral sqrt(1 - D(x)^2) * sqrt(3 + D(x)^2) dx.
(4) D(x) = cos( Integral sqrt(3 + D(x)^2) dx ).
(5.a) D(x) = sqrt(1 - 4*sn(x,2)^2).
(5.b) D(x) = sqrt(4*cn(x,2)^2 - 3).
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).
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

cp's OEIS Frontend

A370543 Expansion of the Jacobi elliptic function cn(x,k) at k = 2 (even powers only).

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