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A279834
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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.
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4
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1, 3, 9, 212, 774, 2986491, 11962183, 5866732236, 24717532254, 155049859325162, 8766713183100126, 1242400321151564076, 157798597956508868, 141417442289739551841, 3032690837599386922473477, 272243517649610491264579553148, 1244664961615535298800024043306
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OFFSET
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0,2
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COMMENTS
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Denominators are given by A279835.
The g.f. A(x) of this sequence equals the square of the g.f. of A279832.
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LINKS
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Paul D. Hanna, Table of n, a(n) for n = 0..50
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FORMULA
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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).
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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 + 3/5*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).
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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-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]), ", ") ); }
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CROSSREFS
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Cf. A279835, A279832, A279833, A060628.
Sequence in context: A174603 A062228 A196864 * A091409 A027891 A073889
Adjacent sequences: A279831 A279832 A279833 * A279835 A279836 A279837
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Dec 26 2016
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STATUS
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approved
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