0,5

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.

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.

J. Tannery and J. Molk, El\'{e}ments de la Th\'{e}orie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.

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).

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! + ...

[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], ...;

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)

Row sums give A000182. Leading terms give A004005.

Cf. A002753.

Sequence in context: A144441 A157150 A142461 this_sequence A022177 A015133 A040202

Adjacent sequences: A060625 A060626 A060627 this_sequence A060629 A060630 A060631

easy,nonn,tabl,nice,new

Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 13 2001