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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. 4
1, 3, 9, 212, 774, 2986491, 11962183, 5866732236, 24717532254, 155049859325162, 8766713183100126, 1242400321151564076, 157798597956508868, 141417442289739551841, 3032690837599386922473477, 272243517649610491264579553148, 1244664961615535298800024043306 (list; graph; refs; listen; history; text; internal format)
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.

Sequence in context: A174603 A062228 A196864 * A091409 A027891 A073889

Adjacent sequences:  A279831 A279832 A279833 * A279835 A279836 A279837

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 26 2016

STATUS

approved

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Last modified December 13 06:26 EST 2019. Contains 329968 sequences. (Running on oeis4.)