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A060627
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1+Sum_{n >= 1} Sum_{k = 0..n-1} (-1)^n*T(n,k)*y^(2*k)*x^(2*n)/(2*n)! = JacobiCN(x,y).
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6
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1, 1, 4, 1, 44, 16, 1, 408, 912, 64, 1, 3688, 30768, 15808, 256, 1, 33212, 870640, 1538560, 259328, 1024, 1, 298932, 22945056, 106923008, 65008896, 4180992, 4096, 1, 2690416, 586629984, 6337665152, 9860488448, 2536974336, 67047424, 16384
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Essentially same triangle as triangle given by [1, 0, 9, 0, 25, 0, 49, 0, 81, 0, 121, ...] DELTA [0, 4, 0, 16, 0, 36, 0, 64, 0, 100, ...] where DELTA is the operator defined in A084938 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 13 2004
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REFERENCES
| CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.20).
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LINKS
| F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]
Eric W. Weisstein, Jacobi Elliptic Functions
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FORMULA
| JacobiCN(x, y)=1 - 1/2*x^2 + (1/24 + 1/6*y^2)*x^4 + ( - 1/720 - 11/180*y^2 - 1/45*y^4)*x^6 + (1/40320 + 17/1680*y^2 + 19/840*y^4 + 1/630*y^6)*x^8 + ( - 1/3628800 - 247/56700*y^6 - 461/453600*y^2 - 641/75600*y^4 - 1/14175*y^8)*x^10 + O(x^12).
From Peter Bala Aug 23 2011
The Taylor expansion of the Jacobian elliptic function cn(x,k)
begins
cn(x,k) = 1-x^2/2!+(1+4*k^2)*x^4/4!-(1+44*k^2+16*k^4)*x^6/6!+....
The coefficient polynomials in this expansion can be calculated
using nested derivatives as follows (see [Dominici, Theorem 4.1 and
Example 4.5]):
Let f(x) = sqrt(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.
Then the coefficient polynomial R(2*n,k) of x^(2*n)/(2*n)! in the
expansion of cn(x,k) is given by 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). See A181613 for the expansion of the reciprocal
function 1/cn(x,k).
[End]
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EXAMPLE
| [1], [1, 4], [1, 44, 16], [1, 408, 912, 64], [1, 3688, 30768, 15808, 256], [1, 33212, 870640, 1538560, 259328, 1024], [1, 298932, 22945056, 106923008, 65008896, 4180992, 4096], [1, 2690416, 586629984, 6337665152, 9860488448, 2536974336, 67047424, 16384], ...
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MAPLE
| A060627 := proc(n, m) JacobiCN(z, k) ; coeftayl(%, z=0, 2*n) ; (-1)^n*coeftayl(%, k=0, 2*m)*(2*n)! ; end proc: # R. J. Mathar, Jan 30 2011
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CROSSREFS
| Row sums: A000364.
Cf. A002754, A060628, A145271, A181612, A181613.
Sequence in context: A169656 A193962 A092667 * A113101 A113112 A069740
Adjacent sequences: A060624 A060625 A060626 * A060628 A060629 A060630
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 13 2001
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