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A279832 Numerators of the coefficients in g.f. A(x) such that: sn(x,i*A(x)) = x, where i^2 = -1, and sn(x,k) is a Jacobi elliptic function. 4
1, 3, 27, 1129, 6177, 228496227, 507769159, 3411091723167, 226108446954939, 2799063804718849119, 56928279095622876861, 175898907783132547767087, 2387767743416733035533529, 617528637834242429324813087883, 26943941094191660800993918030539, 4813884370789026772162811298692933153, 41249694296981783922826921997571040581, 69502372123801730691426662081268221528029689, 19796290340432197210800092215751765052273983, 16957540878135184586375745347497078257426299617863, 1168637136489375278109169401471800538288143908488069 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Denominators are given by A279833.

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^2), A/sqrt(1+A^2) ) = x*sqrt(1+A^2), where sd(x,k) = sn(x,k)/dn(x,k) is a Jacobi elliptic function.

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

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

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

EXAMPLE

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

A(x) = 1 + 3/10*x^2 + 27/200*x^4 + 1129/14000*x^6 + 6177/112000*x^8 + 228496227/5605600000*x^10 + 507769159/16016000000*x^12 + 3411091723167/133413280000000*x^14 + 226108446954939/10673062400000000*x^16 + 2799063804718849119/156146902912000000000*x^18 + 56928279095622876861/3690744977920000000000*x^20 + 175898907783132547767087/13072618711792640000000000*x^22 + 2387767743416733035533529/201117210950656000000000000*x^24 + 617528637834242429324813087883/58382315166865930240000000000000*x^26 + 26943941094191660800993918030539/2835712450962059468800000000000000*x^28 + 4813884370789026772162811298692933153/559968137691477883303936000000000000000*x^30 +...

satisfies: sn(x,i*A(x)) = x.

RELATED SERIES.

The Jacobi elliptic function sn(x,k) begins:

sn(x,k) = x - (k^2 + 1)*x^3/3! + (k^4 + 14*k^2 + 1)*x^5/5! - (k^6 + 135*k^4 + 135*k^2 + 1)*x^7/7! + (k^8 + 1228*k^6 + 5478*k^4 + 1228*k^2 + 1)*x^9/9! - (k^10 + 11069*k^8 + 165826*k^6 + 165826*k^4 + 11069*k^2 + 1)*x^11/11! + (k^12 + 99642*k^10 + 4494351*k^8 + 13180268*k^6 + 4494351*k^4 + 99642*k^2 + 1)*x^13/13! - (k^14 + 896803*k^12 + 116294673*k^10 + 834687179*k^8 + 834687179*k^6 + 116294673*k^4 + 896803*k^2 + 1)*x^15/15! +...

which equals x when k = i*A(x).

A real transformation of the imaginary modulus i*A(x) yields the series:

A(x)/sqrt(1 + A(x)^2) = sqrt(1/2)*(1 + 3/20*x^2 + 27/800*x^4 + 1681/112000*x^6 + 11667/1280000*x^8 + 45274443/7175168000*x^10 + 613581239/130457600000*x^12 + 62857335822759/17076899840000000*x^14 + 8148919947718779/2732303974400000000*x^16 + 198293692034112113343/79947214290944000000000*x^18 + 4605729854262557732997/2188029022699520000000000*x^20 + 243052910628213000290505027/133863615608756633600000000000*x^22 + 38893821159628323146146353/24505925054234624000000000000*x^24 +...).

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

sn(2*x, i*A(x)) = 2*x - 12/5*x^5 - 36/25*x^7 + 1332/875*x^9 + 9984/4375*x^11 - 5136624/21896875*x^13 - 266818932/109484375*x^15 - 77131141044/65143203125*x^17 + 33379542432/19159765625*x^19 + 304830773316936/140153685546875*x^21 - 77528188053360024/154869822529296875*x^23 - 145014068636962776/58668332769921875*x^25 +...

The series y = sn(x/2, i*A(x)) satisfies:

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

where the series y begins:

sn(x/2, i*A(x)) = 1/2*x + 3/320*x^5 + 9/1600*x^7 + 14013/3584000*x^9 + 3729/1280000*x^11 + 6533718813/2870067200000*x^13 + 2402215119/1304576000000*x^15 + 1670885671753959/1092921589760000000*x^17 + 252839176306947/195164569600000000*x^19 + 1427498770243103841051/1279155428655104000000000*x^21 + 4263718777800583142667/4376058045399040000000000*x^23 + 147404533631490298403307261/171345427979208491008000000000*x^25 +...

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-k^2*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] = 3*polcoeff(x - subst(SN, k, I*Ser(A)), #A+2);

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

CROSSREFS

Cf. A279833, A279834, A279835, A060628.

Sequence in context: A331439 A085521 A009066 * A012012 A229866 A305843

Adjacent sequences:  A279829 A279830 A279831 * A279833 A279834 A279835

KEYWORD

nonn,frac

AUTHOR

Paul D. Hanna, Dec 21 2016

STATUS

approved

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Last modified September 20 11:42 EDT 2021. Contains 347584 sequences. (Running on oeis4.)