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)!.

1, 5, 4, 61, 76, 16, 1385, 2424, 1104, 64, 50521, 113672, 79728, 16832, 256, 2702765, 7432604, 7052528, 2586112, 264448, 1024, 199360981, 647923188, 775638816, 408850432, 85975296, 4205568, 4096, 19391512145, 72718170544, 105138354912, 72490884224, 23551644928, 2939602944, 67162112, 16384

Offset

1,2

Comments

The column m=0 is apparently A000364.

References

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover. Section 16.22.

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Links

Table of n, a(n) for n=1..36.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972

P. Bala, A triangle for calculating A181613

D. Dominici, Nested derivatives: A simple method for Computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]

NIST Digital Library of Mathematical Functions, NIST Handbook of Mathematical Functions, Chapter 22

Eric W. Weisstein, Jacobi Elliptic Functions

Formula

From Peter Bala, Aug 23 2011: (Start)

The Taylor expansion of the Jacobian elliptic function cn(u,k) begins

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.

The Taylor expansion of the reciprocal function 1/cn(u,k) can be obtained directly from this by using Jacobi's imaginary transformation

1/cn(u,k) = cn(i*u,sqrt(1-k^2)) [Abramowitz and Stegun, 16.20] to yield

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!+....

The coefficient polynomials R(2*n,k) of this expansion can be calculated as follows (apply [Dominici, Theorem 4.1]):

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).

See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).

(End)

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).

Example

The triangle starts in row n=1 as:
1;
5, 4;
61, 76, 16;
1385, 2424, 1104, 64;
50521, 113672, 79728, 16832, 256;

Maple

A181613 := proc(n, m) JacobiNC(z, k) ; coeftayl(%, z=0, 2*n) ; (-1)^m*coeftayl(%, k=0, 2*m)*(2*n)! ; end proc:
seq( seq(A181613(n, m), m=0..n-1), n=1..10) ;

Mathematica

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 *)

Crossrefs

Cf. A060627, A060628, A181612.

Sequence in context: A262421 A305170 A192344 * A264880 A128191 A186639

Adjacent sequences: A181610 A181611 A181612 * A181614 A181615 A181616

Keyword

tabl,nonn

Author

R. J. Mathar, Jan 30 2011

Status

approved