A279834 Numerators 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, 3, 9, 212, 774, 2986491, 11962183, 5866732236, 24717532254, 155049859325162, 8766713183100126, 1242400321151564076, 157798597956508868, 141417442289739551841, 3032690837599386922473477, 272243517649610491264579553148, 1244664961615535298800024043306

Offset: 0

Author

Paul D. Hanna, Dec 26 2016

Keywords

• nonn

Comments

Denominators are given by A279835.

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

Examples

			This sequence gives the numerators 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. A279835, 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( numerator(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).