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 A279834 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. 4
 1, 3, 9, 212, 774, 2986491, 11962183, 5866732236, 24717532254, 155049859325162, 8766713183100126, 1242400321151564076, 157798597956508868, 141417442289739551841, 3032690837599386922473477, 272243517649610491264579553148, 1244664961615535298800024043306 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Denominators are given by A279835. 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 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). 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( numerator(A[#A]), ", ") ); } CROSSREFS Cf. A279835, A279832, A279833, A060628. Sequence in context: A174603 A062228 A196864 * A091409 A027891 A073889 Adjacent sequences: A279831 A279832 A279833 * A279835 A279836 A279837 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 26 2016 STATUS approved

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Last modified March 24 12:09 EDT 2023. Contains 361479 sequences. (Running on oeis4.)