A370544 -id:A370544 - 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

A370542 Expansion of the Jacobi elliptic function sn(x,k) at k = 2 (odd powers only).

Original entry on oeis.org

1, -5, 73, -2765, 171409, -16145045, 2168436697, -391723265885, 91633164775201, -26955095234906405, 9737498127795037033, -4237907290209405609965, 2187044171819241257792689, -1320533769141977996485790645, 922274662722967857470247551737, -737730926392606318468533810754685

Offset: 0

Paul D. Hanna, Mar 23 2024

sign

			E.g.f.: S(x) = x - 5*x^3/3! + 73*x^5/5! - 2765*x^7/7! + 171409*x^9/9! - 16145045*x^11/11! + 2168436697*x^13/13! - 391723265885*x^15/15! + ...
where S(x) = sn(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. A000182 (unsigned sn(x,1)), A060628 (sn(x,k)).

Cf. A370543 (cn(x,2)), A370544 (dn(x,2)), A249282.

# a(n) = (2*n+1)! * [x^(2*n+1)] sn(x, 2).
sn_list := proc(k, len) local n; seq((2*n+1)!*coeff(series(JacobiSN(z, k), z,
2*len + 2), z, 2*n + 1), n = 0..len) end:
sn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiSN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax] + 1)! (* Jean-François Alcover, Mar 28 2024 *)

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

a(n) = (-1)^n * Sum_{k=0..n} A060628(n,k)*4^k for n >= 0.
E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) S(x) = sn(x,k) at k = 2.
(2.a) S(x) = sn(2*x,1/2)/2.
(2.b) S(x) = sn(x,1/2) * cn(x,1/2) * dn(x,1/2) / (1 - sn(x,1/2)^4/4).
(3.a) S(x) = Series_Reversion( Integral 1/sqrt( (1-x^2)*(1-4*x^2) ) dx ).
(3.b) S(x) = Integral sqrt(1 - S(x)^2) * sqrt(1 - 4*S(x)^2) dx.
(4.a) S(x) = sin( Integral sqrt(1 - 4*S(x)^2) dx ).
(4.b) S(x) = sin( 2 * Integral sqrt(1 - S(x)^2) dx ) / 2.
(5.a) S(x) = sqrt(1 - cn(x,2)^2).
(5.b) S(x) = sqrt(1 - dn(x,2)^2) / 2.
O.g.f.: x/(1 + 5*x - 4*1*2^2*3*x^2/(1 + 5*3^2*x - 4*3*4^2*5*x^2/(1 + 5*5^2*x - 4*5*6^2*7*x^2/(1 + 5*7^2*x - 4*7*8^2*9*x^2/(1 + 5*9^2*x - ...))))) = x - 5*x^2 + 73*x^3 - 2765*x^4 + 171409*x^5 - 16145045*x^6 + ... (continued fraction, see Wall, 94.17, p. 374).
a(n) ~ (-1)^n * 2^(4*n+4) * agm(1,2)^(2*n+2) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * 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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