A171660 Triangle T(n,m) of the expansion coefficients of JacobiCN(x,y) + JacobiDN(x,y) = Sum_{n>=0} Sum_{k=0..n} (-1)^n*T(n,m)*x^(2*n)*y^(2*m)/(2*n)!.

2, 1, 1, 1, 8, 1, 1, 60, 60, 1, 1, 472, 1824, 472, 1, 1, 3944, 46576, 46576, 3944, 1, 1, 34236, 1129968, 3077120, 1129968, 34236, 1, 1, 303028, 27126048, 171931904, 171931904, 27126048, 303028, 1, 1, 2706800, 653677408, 8874639488, 19720976896

Offset: 0

Author

Roger L. Bagula, Dec 14 2009

Keywords

• nonn
• tabl

Comments

Row sums are 2*A000364(n).

Since the coefficients of JacobiCN are in A060627 and the coefficients of JacobiDN are obtained by row-reversal of A060627, this triangle here is a symmetrized variant, adding A060627 and its mirrored version.

Examples

			2;
1, 1;
1, 8, 1;
1, 60, 60, 1;
1, 472, 1824, 472, 1;
1, 3944, 46576, 46576, 3944, 1;
1, 34236, 1129968, 3077120, 1129968, 34236, 1;
1, 303028, 27126048, 171931904, 171931904, 27126048, 303028, 1;
1, 2706800, 653677408, 8874639488, 19720976896, 8874639488, 653677408, 2706800, 1;
1, 24279312, 15877769376, 440712200064, 1948265426688, 1948265426688, 440712200064, 15877769376, 24279312, 1;
1, 218186164, 388726995744, 21489645169920, 176743676925696, 343497841920000, 176743676925696, 21489645169920, 388726995744, 218186164, 1;
		

Crossrefs

Cf. A060627.

Programs

Maple

A171660 := proc(n,m) JacobiCN(z,k) +JacobiDN(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^n*coeftayl(%,k=0,2*m)*(2*n)! ; end proc: # R. J. Mathar, Jan 30 2011

Mathematica

p[t_] = JacobiCN[t, x] + JacobiDN[t, x]
a = Table[ CoefficientList[FullSimplify[ExpandAll[(-1)^Floor[n/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 20, 2}]
Flatten[a]