A279835 Denominators of the coefficients in g.f. A(x) such that: sn(x,-A(x)) = x, where sn(x,m) is a Jacobi elliptic function.

1, 5, 25, 875, 4375, 21896875, 109484375, 65143203125, 325716015625, 2382612654296875, 154869822529296875, 24934041427216796875, 3562005918173828125, 3559956170522705078125, 84510816662372930908203125, 8344175483159391333221435546875, 41720877415796956666107177734375, 11291964076972525306465238189697265625

Offset: 0

Author

Paul D. Hanna, Dec 26 2016

Keywords

• nonn

Comments

Numerators are given by A279834.

The g.f. A(x) of this sequence equals the square of the g.f. of A279832.

Examples

			This sequence gives the denominators of the coefficients in g.f. A(x), such that
A(x) = 1 + 3/5*x^2 + 9/25*x^4 + 212/875*x^6 + 774/4375*x^8 + 2986491/21896875*x^10 + 11962183/109484375*x^12 + 5866732236/65143203125*x^14 + 24717532254/325716015625*x^16 + 155049859325162/2382612654296875*x^18 + 8766713183100126/154869822529296875*x^20 + 1242400321151564076/24934041427216796875*x^22 + 157798597956508868/3562005918173828125*x^24 + 141417442289739551841/3559956170522705078125*x^26 + 3032690837599386922473477/84510816662372930908203125*x^28 + 272243517649610491264579553148/8344175483159391333221435546875*x^30 + 1244664961615535298800024043306/41720877415796956666107177734375*x^32 + 309586737719752481090144972054844018/11291964076972525306465238189697265625*x^34 + 1428965605601484765267196303905398982/56459820384862626532326190948486328125*x^36 + 1900644020251253780726568413610042774696/81019842252277869073888084011077880859375*x^38 + 10448090522732112432951611797351884498204/478753613308914680891156860065460205078125*x^40 +...
satisfies: sn(x,-A(x)) = x.
RELATED SERIES.
The Jacobi elliptic function sn(x,m) begins:
sn(x,m) = x - (m + 1)*x^3/3! + (m^2 + 14*m + 1)*x^5/5! - (m^3 + 135*m^2 + 135*m + 1)*x^7/7! + (m^4 + 1228*m^3 + 5478*m^2 + 1228*m + 1)*x^9/9! - (m^5 + 11069*m^4 + 165826*m^3 + 165826*m^2 + 11069*m + 1)*x^11/11! + (m^6 + 99642*m^5 + 4494351*m^4 + 13180268*m^3 + 4494351*m^2 + 99642*m + 1)*x^13/13! - (m^7 + 896803*m^6 + 116294673*m^5 + 834687179*m^4 + 834687179*m^3 + 116294673*m^2 + 896803*m + 1)*x^15/15! +...
which equals x when m = -A(x).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A279834, A279832, A279833, A060628.

Programs

PARI

/* Code to list first N nonzero terms of vector A: */ {N=20;
/* Generate 2*N terms of Jacobi Elliptic Function SN: */
SN = serreverse(intformal(1/sqrt((1-x^2)*(1-m*x^2) +x*O(x^(2*N+2))) ));
/* Print N terms of this sequence: */
A=[1]; print1(A[1], ", ");
for(i=1, N, A = concat(A, [0, 0]);
A[#A] = 6*polcoeff(x - subst(SN, m, -Ser(A)), #A+2);
print1( denominator(A[#A]), ", ") ); }

Formula

G.f. A = A(x) satisfies:

(1) sd( x*sqrt(1+A), A/(1+A) ) = x*sqrt(1+A), where sd(x,m) = sn(x,m)/dn(x,m) is a Jacobi elliptic function.

(2) sn(2*x, -A(x)) = 2*x*sqrt(1-x^2)*sqrt(1 + x^2*A)/(1 + x^4*A).

(3) y = sn(x/2, -A(x)) is a solution to the equation:

x^2*(1 + A*y^4)^2 = 4*y^2*(1-y^2)*(1 + A*y^2).