cp's OEIS Frontend

A181613 Triangle T(n,m) of the coefficients JacobiNC(x,y) = sum_{n>0} sum_{m=0..n-1} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!.

%I A181613 #22 Mar 31 2025 12:03:01
%S A181613 1,5,4,61,76,16,1385,2424,1104,64,50521,113672,79728,16832,256,
%T A181613 2702765,7432604,7052528,2586112,264448,1024,199360981,647923188,
%U A181613 775638816,408850432,85975296,4205568,4096,19391512145,72718170544,105138354912,72490884224,23551644928,2939602944,67162112,16384
%N A181613 Triangle T(n,m) of the coefficients JacobiNC(x,y) = sum_{n>0} sum_{m=0..n-1} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!.
%C A181613 The column m=0 is apparently A000364.
%D A181613 M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover. Section 16.22.
%D A181613 H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.
%H A181613 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.
%H A181613 P. Bala, <a href="/A181613/a181613.pdf">A triangle for calculating A181613</a>
%H A181613 D. Dominici, <a href="https://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for Computing series expansions of inverse functions</a>, arXiv:math/0501052v2 [math.CA].
%H A181613 <a href="https://dlmf.nist.gov/">NIST Digital Library of Mathematical Functions</a>, NIST Handbook of Mathematical Functions, Chapter 22.
%H A181613 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/JacobiEllipticFunctions.html">Jacobi Elliptic Functions</a>.
%F A181613 From _Peter Bala_, Aug 23 2011: (Start)
%F A181613 The Taylor expansion of the Jacobian elliptic function cn(u,k) begins
%F A181613 cn(u,k) = 1-u^2/2!+(1+4*k^2)*u^4/4!-(1+44*k^2+16*k^4)*u^6/6!+... - see A060627.
%F A181613 The Taylor expansion of the reciprocal function 1/cn(u,k) can be obtained directly from this by using Jacobi's imaginary transformation
%F A181613 1/cn(u,k) = cn(i*u,sqrt(1-k^2)) [Abramowitz and Stegun, 16.20] to yield
%F A181613 1/cn(u,k) = 1+u^2/2!+(5-4*k^2)*u^4/4!+(61-76*k^2+16*k^4)*u^6/6!+....
%F A181613 The coefficient polynomials R(2*n,k) of this expansion can be calculated as follows (apply [Dominici, Theorem 4.1]):
%F A181613 Let f(x) = sqrt(k^2-cos^2(x)). 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 R(2*n,k) = D^(2*n)[f](0).
%F A181613 See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
%F A181613 (End)
%F A181613 G.f. 1/(1 - x/(1 - 2^2*(1 - k^2)*x/(1 - 3^2*x/(1 - 4^2*(1 - k^2)*x/(1 - 5^2*x/(1 - ...)))))) = 1 + x + (5 - 4*k^2)*x^2 + (61 - 76*k^2 + 16*k^4)*x^3 + ... (see Wall, 94.19, p. 374).
%e A181613 The triangle starts in row n=1 as:
%e A181613 1;
%e A181613 5, 4;
%e A181613 61, 76, 16;
%e A181613 1385, 2424, 1104, 64;
%e A181613 50521, 113672, 79728, 16832, 256;
%p A181613 A181613 := proc(n,m) JacobiNC(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^m*coeftayl(%,k=0,2*m)*(2*n)! ; end proc:
%p A181613 seq( seq(A181613(n,m),m=0..n-1),n=1..10) ;
%t A181613 nmax = 8; se = Series[JacobiNC[x, y], {x, 0, 2*nmax}]; t[n_, m_] := Coefficient[se, x, 2*n]*(2*n)! // Coefficient[#, y, m]& // Abs; Table[t[n, m], {n, 1, nmax}, {m, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jan 10 2014 *)
%Y A181613 Cf. A060627, A060628, A181612.
%K A181613 tabl,nonn
%O A181613 1,2
%A A181613 _R. J. Mathar_, Jan 30 2011