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A279832
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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.
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4
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1, 3, 27, 1129, 6177, 228496227, 507769159, 3411091723167, 226108446954939, 2799063804718849119, 56928279095622876861, 175898907783132547767087, 2387767743416733035533529, 617528637834242429324813087883, 26943941094191660800993918030539, 4813884370789026772162811298692933153, 41249694296981783922826921997571040581, 69502372123801730691426662081268221528029689, 19796290340432197210800092215751765052273983, 16957540878135184586375745347497078257426299617863, 1168637136489375278109169401471800538288143908488069
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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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).
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EXAMPLE
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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 +...
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PROG
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(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]), ", ") ); }
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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