cp's OEIS Frontend

A181612 Triangle T(n,m) of the coefficients JacobiDC(x,y) = sum_{n>=0} sum_{m=0..n} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!.

%I A181612 #23 Mar 31 2025 11:58:45
%S A181612 1,1,1,5,6,1,61,107,47,1,1385,3116,2142,412,1,50521,138933,130250,
%T A181612 45530,3693,1,2702765,8783986,10430983,5353260,1036715,33218,1,
%U A181612 199360981,747603679,1074680289,728130163,226132303
%N A181612 Triangle T(n,m) of the coefficients JacobiDC(x,y) = sum_{n>=0} sum_{m=0..n} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!.
%D A181612 M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover. Section 16.22.
%H A181612 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 A181612 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 A181612 <a href="https://dlmf.nist.gov/">NIST Digital Library of Mathematical Functions</a>, NIST Handbook of Mathematical Functions, Chapter 22.
%H A181612 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/JacobiEllipticFunctions.html">Jacobi Elliptic Functions</a>.
%F A181612 From Peter Bala, Aug 23 2011: (Start)
%F A181612 The elliptic function dc(x,k) (JacobiDC(x,k) in Maple notation) is defined as dn(x,k)/cn(x,k) where dn(x,k) and cn(x,k) are the Jacobian elliptic functions of modulus k. The Taylor expansions begin
%F A181612 dn(x,k) = 1-k^2*x^2/2!+k^2*(4+k^2)*x^4/4!-k^2*(16+44*k^2+k^4)*x^6/6!+...
%F A181612 cn(x,k) = 1-x^2/2!+(1+4*k^2)*x^4/4!-(1+44*k^2+16*k^4)*x^6/6!+... and hence
%F A181612 dc(x,k) = 1+(1-k^2)*x^2/2!+(5-6*k^2+k^4)*x^4/4!+(61-107*k^2+47*k^4-k^6)*x^6/6!+....
%F A181612 The coefficients for cn(x,k) are in A060627. The coefficients of dn(x,k) may be obtained by row reversal of A060627.
%F A181612 The expansion for dc(x,k) can also be obtained directly from that of dn(x,k) since by Jacobi's imaginary transformations we have dc(x,k) = dn(i*x,k'), where the complementary modulus k' is given by k' = sqrt(1-k^2).
%F A181612 By Jacobi's real transformation the reciprocal of dc(x,k) is given by 1/dc(x,k) = dc(x*k,1/k).
%F A181612 The row polynomials of this table can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1 and Example 4.5]):
%F A181612 Let f(x) = sqrt(1-(1-k^2)*sin^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.
%F A181612 See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
%F A181612 Then the coefficient of x^(2*n)/(2*n)! in the expansion of dc(x,k) is given by (-1)^n*D^(2*n)[f](0).
%F A181612 (End)
%e A181612 The triangle starts in row n=0 as
%e A181612 1;
%e A181612 1, 1;
%e A181612 5, 6, 1;
%e A181612 61, 107, 47, 1;
%e A181612 1385, 3116, 2142, 412, 1;
%e A181612 50521, 138933, 130250, 45530, 3693, 1;
%p A181612 A181612 := proc(n,m) JacobiDC(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^m*coeftayl(%,k=0,2*m)*(2*n)! ; end proc:
%p A181612 seq( seq(A181612(n,m),m=0..n),n=0..10) ;
%t A181612 nmax = 8; se = Series[JacobiDC[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, 0, nmax}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 09 2014 *)
%Y A181612 Cf. A060627, A060628, A181613, A000364 (apparently the column m=0).
%K A181612 nonn,tabl
%O A181612 0,4
%A A181612 _R. J. Mathar_, Jan 30 2011