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A181612
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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)!.
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3
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1, 1, 1, 5, 6, 1, 61, 107, 47, 1, 1385, 3116, 2142, 412, 1, 50521, 138933, 130250, 45530, 3693, 1, 2702765, 8783986, 10430983, 5353260, 1036715, 33218, 1, 199360981, 747603679, 1074680289, 728130163, 226132303
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OFFSET
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0,4
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REFERENCES
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M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover. Section 16.22.
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LINKS
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FORMULA
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From Peter Bala, Aug 23 2011: (Start)
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
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!+...
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
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!+....
The coefficients for cn(x,k) are in A060627. The coefficients of dn(x,k) may be obtained by row reversal of A060627.
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).
By Jacobi's real transformation the reciprocal of dc(x,k) is given by 1/dc(x,k) = dc(x*k,1/k).
The row polynomials of this table can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1 and Example 4.5]):
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.
See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
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).
(End)
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EXAMPLE
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The triangle starts in row n=0 as
1;
1, 1;
5, 6, 1;
61, 107, 47, 1;
1385, 3116, 2142, 412, 1;
50521, 138933, 130250, 45530, 3693, 1;
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MAPLE
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A181612 := proc(n, m) JacobiDC(z, k) ; coeftayl(%, z=0, 2*n) ; (-1)^m*coeftayl(%, k=0, 2*m)*(2*n)! ; end proc:
seq( seq(A181612(n, m), m=0..n), n=0..10) ;
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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