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 A279835 Denominators 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, 5, 25, 875, 4375, 21896875, 109484375, 65143203125, 325716015625, 2382612654296875, 154869822529296875, 24934041427216796875, 3562005918173828125, 3559956170522705078125, 84510816662372930908203125, 8344175483159391333221435546875, 41720877415796956666107177734375, 11291964076972525306465238189697265625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Numerators are given by A279834. 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 denominators 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( denominator(A[#A]), ", ") ); } CROSSREFS Cf. A279834, A279832, A279833, A060628. Sequence in context: A218150 A176594 A274463 * A169652 A359232 A359235 Adjacent sequences: A279832 A279833 A279834 * A279836 A279837 A279838 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 26 2016 STATUS approved

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Last modified March 31 03:00 EDT 2023. Contains 361626 sequences. (Running on oeis4.)