A279834 - OEIS

%I #14 Mar 12 2017 16:47:21

%S 1,3,9,212,774,2986491,11962183,5866732236,24717532254,

%T 155049859325162,8766713183100126,1242400321151564076,

%U 157798597956508868,141417442289739551841,3032690837599386922473477,272243517649610491264579553148,1244664961615535298800024043306

%N 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.

%C Denominators are given by A279835.

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

%H Paul D. Hanna, <a href="/A279834/b279834.txt">Table of n, a(n) for n = 0..50</a>

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

%F (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.

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

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

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

%e This sequence gives the numerators of the coefficients in g.f. A(x), such that

%e 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 +...

%e satisfies: sn(x,-A(x)) = x.

%e RELATED SERIES.

%e The Jacobi elliptic function sn(x,m) begins:

%e 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! +...

%e which equals x when m = -A(x).

%o (PARI) /* Code to list first N nonzero terms of vector A: */ {N=20;

%o /* Generate 2*N terms of Jacobi Elliptic Function SN: */

%o SN = serreverse(intformal(1/sqrt((1-x^2)*(1-m*x^2) +x*O(x^(2*N+2))) ));

%o /* Print N terms of this sequence: */

%o A=[1]; print1(A[1], ", ");

%o for(i=1, N, A = concat(A, [0, 0]);

%o A[#A] = 6*polcoeff(x - subst(SN, m, -Ser(A)), #A+2);

%o print1( numerator(A[#A]), ", ") ); }

%Y Cf. A279835, A279832, A279833, A060628.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 26 2016