This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060628 #57 Mar 31 2025 11:56:53
%S A060628 1,1,1,1,14,1,1,135,135,1,1,1228,5478,1228,1,1,11069,165826,165826,
%T A060628 11069,1,1,99642,4494351,13180268,4494351,99642,1,1,896803,116294673,
%U A060628 834687179,834687179,116294673,896803,1,1,8071256,2949965020,47152124264,109645021894,47152124264,2949965020,8071256,1
%N A060628 Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.
%D A060628 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
%D A060628 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.24).
%H A060628 M. Abramowitz and I. A. Stegun, eds., <a href="https://personal.math.ubc.ca/~cbm/aands/toc.htm">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.22.1.
%H A060628 A. Cayley, <a href="https://newcatalog.library.cornell.edu/catalog/613744">An Elementary Treatise on Elliptic Functions</a> (page images), G. Bell and Sons, London, 1895, p. 56.
%H A060628 F. Clarke, <a href="http://www-maths.swan.ac.uk/staff/fwc/Gregynog-slides.pdf">The Taylor Series Coefficients of the Jacobi Elliptic Functions</a>, slides. [broken link]
%H A060628 D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions.</a> arXiv:math/0501052v2 [math.CA], 2005.
%H A060628 A. Fransen, <a href="http://dx.doi.org/10.1090/S0025-5718-1981-0628708-X">Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k)</a>, Math. Comp., 37 (1981), 475-497.
%H A060628 R. Fricke, <a href="http://dx.doi.org/10.1007/978-3-642-19557-0_8">Die elliptischen Funktionen und ihre Anwendungen, Erster Teil</a>, p. 399 with p. 397.
%H A060628 C. L. Mallows, <a href="/A004004/a004004.pdf">Letter to N. J. A. Sloane, May 16 1973</a>
%H A060628 J. Tannery and J. Molk, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k99586x/f104.image">Eléments de la Théorie des Fonctions Elliptiques (Vol. 4)</a>, Gauthier-Villars, Paris, 1902, p. 92.
%H A060628 G. Viennot, <a href="http://dx.doi.org/10.1016/0097-3165(80)90001-1">Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi</a>, J. Combin. Theory, A 29 (1980), 121-133.
%H A060628 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiEllipticFunctions.html">Jacobi Elliptic Functions</a>
%F A060628 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).
%F A060628 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).
%F A060628 From _Peter Bala_, Aug 23 2011: (Start)
%F A060628 Let f(x) = sqrt((1-x^2)*(1-k^2*x^2)).
%F A060628 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.
%F A060628 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].
%F A060628 See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
%F A060628 (End)
%F A060628 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
%e A060628 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! + ...
%e A060628 The triangle T(n, m) begins:
%e A060628 n\m 0 1 2 3 4 5 6 7
%e A060628 0: 1
%e A060628 2: 1 1
%e A060628 3: 1 14 1
%e A060628 4: 1 135 135 1
%e A060628 5: 1 1228 5478 1228 1
%e A060628 6: 1 11069 165826 165826 11069 1
%e A060628 7: 1 99642 4494351 13180268 4494351 99642 1
%e A060628 8: 1 896803 116294673 834687179 834687179 116294673 896803 1
%e A060628 ... reformatted. - _Wolfdieter Lang_, Jul 05 2016
%p A060628 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.
%p A060628 "SN ", 1 "CN ", 1 "DN ", 1
%p A060628 "SN ", 1, 1 "CN ", 1, 4 "DN ", 4, 1
%p A060628 "SN ", 1, 14, 1 "CN ", 1, 44, 16 "DN ", 16, 44, 1
%p A060628 "SN ", 1, 135, 135, 1 "CN ", 1, 408, 912, 64 "DN ", 64, 912, 408, 1
%p A060628 "SN ", 1, 1228, 5478, 1228, 1 "CN ", 1, 3688, 30768, 15808, 256 "DN ", 256, 15808, 30768, 3688, 1
%p A060628 "SN ", 1, 11069, 165826, 165826, 11069, 1 "CN ", 1, 33212, 870640, 1538560, 259328, 1024 "DN ", 1024, 259328, 1538560, 870640, 33212, 1
%p A060628 #----------------------------------------------------------------
%p A060628 # Taylor series coefficients of Jacobi SN,CN,DN
%p A060628 #----------------------------------------------------------------
%p A060628 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);
%p A060628 # 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)))));
%p A060628 # 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)
%p A060628 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
%t A060628 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 *)
%Y A060628 Row sums give A000182. Diagonals: A004004, A004005, A002753, A032348, A032427, A032428, A032429, A032430, A032431, A032432, A032433.
%K A060628 easy,nonn,tabl,nice
%O A060628 0,5
%A A060628 _Vladeta Jovovic_, Apr 13 2001