A279834 - OEIS

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

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OFFSET

0,2

COMMENTS

Denominators are given by A279835.

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

LINKS

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

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).

EXAMPLE

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).

PROG

(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]), ", ") ); }

CROSSREFS

Cf. A279835, A279832, A279833, A060628.