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A060628
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Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.
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7
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1, 1, 1, 1, 14, 1, 1, 135, 135, 1, 1, 1228, 5478, 1228, 1, 1, 11069, 165826, 165826, 11069, 1, 1, 99642, 4494351, 13180268, 4494351, 99642, 1, 1, 896803, 116294673, 834687179, 834687179, 116294673, 896803, 1, 1, 8071256, 2949965020
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OFFSET
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0,5
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REFERENCES
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CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.24).
G. Viennot, Une interpretation combinatoire des coefficients des developpements en serie entiere des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
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LINKS
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Table of n, a(n) for n=0..38.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]
J. Tannery and J. Molk, El\'{e}ments de la Th\'{e}orie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
Eric W. Weisstein, Jacobi Elliptic Functions
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FORMULA
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Sum_{n>=0} Sum_{k=0..n} (-1)^n*T(n, k)*y^(2*k)*x^(2*n+1)/(2*n+1)! = JacobiSN(x, y).
JacobiSN(x, y)=1*x+(-1/6-1/6*y^2)*x^3+(1/120+7/60*y^2+1/120*y^4)*x^5+(-1/5040-3/112*y^4-3/112*y^2-1/5040*y^6)*x^7+(1/362880+307/90720*y^6+913/60480*y^4+307/90720*y^2+1/362880*y^8)*x^9+O(x^11).
From Peter Bala, Aug 23 2011: (Start)
Let f(x) = sqrt((1-x^2)*(1-k^2*x^2)).
Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.
Then the coefficient polynomial R(2*n+1,k) of u^(2*n+1)/(2*n+1)! is given by R(2*n+1,k) = D^(2*n)[f](0) - apply [Dominici, Theorem 4.1].
See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
(End)
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EXAMPLE
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sn u = u - (1 + k^2)u^3/3! + (1 + 14k^2 +k^4)u^5/5! - (1 + 135 k^2 + 135 k^4 + k^6) u^7/7! + ...
Triangle begins
[1],
[1, 1],
[1, 14, 1],
[1, 135, 135, 1],
[1, 1228, 5478, 1228, 1],
[1, 11069, 165826, 165826, 11069, 1],
[1, 99642, 4494351, 13180268, 4494351, 99642, 1], ...;
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MAPLE
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Maple program from Rostislav Kollman (kollman(AT)dynasig.cz), Nov 05 2009 (Start) The program generates an "all in one" triangle of Taylor coefficients of the Jacobi SN, CN, DN functions.
"SN ", 1 "CN ", 1 "DN ", 1
"SN ", 1, 1 "CN ", 1, 4 "DN ", 4, 1
"SN ", 1, 14, 1 "CN ", 1, 44, 16 "DN ", 16, 44, 1
"SN ", 1, 135, 135, 1 "CN ", 1, 408, 912, 64 "DN ", 64, 912, 408, 1
"SN ", 1, 1228, 5478, 1228, 1 "CN ", 1, 3688, 30768, 15808, 256 "DN ", 256, 15808, 30768, 3688, 1
"SN ", 1, 11069, 165826, 165826, 11069, 1 "CN ", 1, 33212, 870640, 1538560, 259328, 1024 "DN ", 1024, 259328, 1538560, 870640, 33212, 1
#----------------------------------------------------------------
# Taylor series coefficients of Jacobi SN, CN, DN
#----------------------------------------------------------------
n := 6: g := x: for i from 1 to 2*n do g := simplify(y*z*diff(g, x) + x*z*diff(g, y) + x*y*diff(g, z)); if(type(i, odd))then SN := simplify(sort(subs(z = k, subs(y = 1, subs(x = 0, g)))) / k);
# lprint("SN ", SN); lprint("SN ", seq(coeff(SN, k, j), j=0..i-1, 2)); else CN := simplify(sort(subs(z = 1, subs(y = 0, subs(x = k, g)))) / k); DN := simplify(sort(subs(z = 0, subs(y = k, subs(x = 1, g)))));
# lprint("CN ", CN); # lprint("DN ", DN); lprint("CN ", seq(coeff(CN, k, j), j=0..i-2, 2)); lprint("DN ", seq(coeff(DN, k, j), j=2..i, 2)); end; end: (End)
A060628 := proc(n, m) JacobiSN(z, k) ; coeftayl(%, z=0, 2*n+1) ; (-1)^n*coeftayl(%, k=0, 2*m)*(2*n+1)! ; end proc: # alternative program, R. J. Mathar, Jan 30 2011
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MATHEMATICA
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maxn = 8; se = Series[ JacobiSN[u, m], {u, 0, 2*maxn + 1 }]; cc = Partition[ CoefficientList[se, u], 2][[All, 2]]; Flatten[ (CoefficientList[#, m] & /@ cc)* Table[(-1)^n*(2*n + 1)!, {n, 0, maxn}]] (* From Jean-François Alcover, Sep 21 2011 *)
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CROSSREFS
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Row sums give A000182. Leading terms give A004005.
Cf. A002753.
Sequence in context: A157150 A142461 A174720 * A022177 A015133 A040202
Adjacent sequences: A060625 A060626 A060627 * A060629 A060630 A060631
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KEYWORD
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easy,nonn,tabl,nice,changed
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AUTHOR
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Vladeta Jovovic, Apr 13 2001
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STATUS
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approved
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