A004005 -id:A004005 - cp's OEIS frontend

A060628 Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.

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, 47152124264, 109645021894, 47152124264, 2949965020, 8071256, 1

Offset: 0

Vladeta Jovovic, Apr 13 2001

			sn u = u - (1 + k^2)*u^3/3! + (1 + 14*k^2 + k^4)*u^5/5! - (1 + 135*k^2 + 135*k^4 + k^6)*u^7/7! + ...
The triangle T(n, m) begins:
n\m 0      1         2         3         4         5      6  7
0:  1
2:  1      1
3:  1     14         1
4:  1    135       135         1
5:  1   1228      5478      1228         1
6:  1  11069    165826    165826     11069         1
7:  1  99642   4494351  13180268   4494351     99642      1
8:  1 896803 116294673 834687179 834687179 116294673 896803  1
... reformatted. - _Wolfdieter Lang_, Jul 05 2016

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.24).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.22.1.
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. [broken link]
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Erster Teil, p. 399 with p. 397.
C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
Eric Weisstein's World of Mathematics, Jacobi Elliptic Functions

Row sums give A000182. Diagonals: A004004, A004005, A002753, A032348, A032427, A032428, A032429, A032430, A032431, A032432, A032433.

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

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}]] (* Jean-François Alcover, Sep 21 2011 *)

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)
sn(u|k^2) = Sum_{n>=0} a_n(k^2)*u^(2*n+1)/(2*n+1)!. For the recurrence of the row polynomials a_n(k^2) = Sum_{m=0..n} (-1)^n*T(n, m)*k^(2*m), see the Fricke reference. - Wolfdieter Lang, Jul 05 2016

A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

Original entry on oeis.org

1, 2, 5, 14, 38, 101, 275, 746, 2026, 5507, 14969, 40689, 110604, 300652, 817255, 2221528, 6038739, 16414993, 44620576, 121291299, 329703934, 896228212, 2436200862, 6622280533, 18001224835, 48932402358, 133012060152, 361564266077, 982833574297, 2671618645410

Offset: 1

Michel Lagneau, Jan 30 2010

nonn

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt and the complete elliptic integral CK is defined by CK(k) = F(1,sqrt(1-k^2)). We calculate the values of CK(k) with k = 1/p, p = 1,2,3, ... and we propose a very interesting property: a(n+1)/a(n) tends toward e = 2.7182818... when n tends to infinity. For example, a(8) / a(7) = 2.718281581; a(9) / a(8) = 2.7182817562.

			a(3) = 38 because floor(CK(1/37)) = 4 and floor(CK(1/38)) = 5.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.
M. Abramowitz and I. Stegun, "Elliptic Integrals", Chapter 17 of Handbook of Mathematical Functions. Dover Publications Inc., New York, 1046 p., (1965).
A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 56.
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]

Cf. elliptic functions: A001936, A002318, A001937, A001934, A001938, A002754, A001939, A001940, A001941, A002753, A006089, A004005.

a0:=1:for p from 1 to 1000 do:a:= evalf(EllipticCK(1/p)):if floor(a)=a0+1 then print(p):a0:=floor(a):else fi:od:

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt. CK is defined by CK(k) = F(1,sqrt(1-k^2)). a(n) is the n-th integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

cp's OEIS Frontend

A060628 Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.

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