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 A153300 Coefficient of x^(4n)/(4n)! in the Maclaurin expansion of cm4(x), which is a generalization of the Dixon elliptic function cm(x,0) defined by A104134. 3
 1, 6, 2268, 7434504, 95227613712, 3354162536029536, 264444869673131894208, 40740588107524550752746624, 11136881432872615930509713801472, 5026062205760019668688216299061782016 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA Define sm4(x)^4 = cm4(x)^4 - 1, where sm4(x) is the g.f. of A153301, then: d/dx cm4(x) = sm4(x)^3 ; d/dx sm4(x) = cm4(x)^3 . EXAMPLE G.f.: cm4(x) = 1 + 6*x^4/4! + 2268*x^8/8! + 7434504*x^12/12! + 95227613712*x^16/16! +... sm4(x) = x + 18*x^5/5! + 14364*x^9/9! + 70203672*x^13/13! + 1192064637456*x^17/17! +... These functions satisfy: cm4(x)^4 - sm4(x)^4 = 1 where: cm4(x)^4 = 1 + 24*x^4/4! + 24192*x^8/8! + 140507136*x^12/12! + 2716743794688*x^16/16 +... RELATED EXPANSIONS: cm4(x)^2 = 1 + 12*x^4/4! + 7056*x^8/8! + 28340928*x^12/12! + 419025809664*x^16/16! +... sm4(x)^2 = 2*x^2/2! + 216*x^6/6! + 368928*x^10/10! + 3000945024*x^14/14! +... cm4(x)^3 = 1 + 18*x^4/4! + 14364*x^8/8! + 70203672*x^12/12! + 1192064637456*x^16/16! +... sm4(x)^3 = 6*x^3/3! + 2268*x^7/7! + 7434504*x^11/11! + 95227613712*x^15/15! +... DERIVATIVES: d/dx cm4(x) = sm4(x)^3 ; d^2/dx^2 cm4(x) = 3*cm4(x)^3*sm4(x)^2 ; d^3/dx^3 cm4(x) = 6*cm4(x)^6*sm4(x) + 9*cm4(x)^2*sm4(x)^5 ; d^4/dx^4 cm4(x) = 6*cm4(x)^9 + 81*cm4(x)^5*sm4(x)^4 + 18*cm4(x)*sm4(x)^8 ;... MATHEMATICA With[{n = 9}, CoefficientList[Series[JacobiDN[Sqrt[2] x^(1/4), 1/2]/Sqrt[JacobiCN[Sqrt[2] x^(1/4), 1/2]], {x, 0, n}], x] Table[(4 k)!, {k, 0, n}]] (* Jan Mangaldan, Jan 04 2017 *) PROG (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x); for(i=0, n, A=1+intformal(intformal(A^3)^3)); n=4*n; n!*polcoeff(A, n))} CROSSREFS Cf. A104134; A153301, A153302 (cm4(x)^2 + sm4(x)^2). Cf. A153303 (cm4(x)^4). [From Paul D. Hanna, Jan 03 2009] Sequence in context: A056048 A051113 A067174 * A059203 A254005 A279654 Adjacent sequences:  A153297 A153298 A153299 * A153301 A153302 A153303 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 02 2009 STATUS approved

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