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A060628
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Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.
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21
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1, 1, 1, 1, 14, 1, 1, 135, 135, 1, 1, 1228, 5478, 1228, 1, 1, 11069, 165826, 165826, 11069, 1, 1, 99642, 4494351, 13180268, 4494351, 99642, 1, 1, 896803, 116294673, 834687179, 834687179, 116294673, 896803, 1, 1, 8071256, 2949965020, 47152124264, 109645021894, 47152124264, 2949965020, 8071256, 1
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OFFSET
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0,5
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REFERENCES
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CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.24).
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LINKS
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FORMULA
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Sum_{n>=0} Sum_{k=0..n} (-1)^n*T(n, k)*y^(2*k)*x^(2*n+1)/(2*n+1)! = JacobiSN(x, y).
JacobiSN(x, y) = 1*x + (-1/6 - (1/6)*y^2)*x^3 + (1/120 + (7/60)*y^2 + (1/120)*y^4)*x^5 + (-1/5040 - (3/112)*y^4 - (3/112)*y^2 - (1/5040)*y^6)*x^7 + (1/362880 + (307/90720)*y^6 + (913/60480)*y^4 + (307/90720)*y^2 + (1/362880)*y^8)*x^9 + O(x^11).
Let f(x) = sqrt((1-x^2)*(1-k^2*x^2)).
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+1,k) of u^(2*n+1)/(2*n+1)! is given by R(2*n+1,k) = D^(2*n)[f](0) - apply [Dominici, Theorem 4.1].
See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
(End)
sn(u|k^2) = Sum_{n>=0} a_n(k^2)*u^(2*n+1)/(2*n+1)!. For the recurrence of the row polynomials a_n(k^2) = Sum_{m=0..n} (-1)^n*T(n, m)*k^(2*m), see the Fricke reference. - Wolfdieter Lang, Jul 05 2016
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EXAMPLE
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sn u = u - (1 + k^2)*u^3/3! + (1 + 14*k^2 + k^4)*u^5/5! - (1 + 135*k^2 + 135*k^4 + k^6)*u^7/7! + ...
The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 6 7
0: 1
2: 1 1
3: 1 14 1
4: 1 135 135 1
5: 1 1228 5478 1228 1
6: 1 11069 165826 165826 11069 1
7: 1 99642 4494351 13180268 4494351 99642 1
8: 1 896803 116294673 834687179 834687179 116294673 896803 1
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MAPLE
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Maple program from Rostislav Kollman (kollman(AT)dynasig.cz), Nov 05 2009: (Start) The program generates an "all in one" triangle of Taylor coefficients of the Jacobi SN, CN, DN functions.
"SN ", 1 "CN ", 1 "DN ", 1
"SN ", 1, 1 "CN ", 1, 4 "DN ", 4, 1
"SN ", 1, 14, 1 "CN ", 1, 44, 16 "DN ", 16, 44, 1
"SN ", 1, 135, 135, 1 "CN ", 1, 408, 912, 64 "DN ", 64, 912, 408, 1
"SN ", 1, 1228, 5478, 1228, 1 "CN ", 1, 3688, 30768, 15808, 256 "DN ", 256, 15808, 30768, 3688, 1
"SN ", 1, 11069, 165826, 165826, 11069, 1 "CN ", 1, 33212, 870640, 1538560, 259328, 1024 "DN ", 1024, 259328, 1538560, 870640, 33212, 1
#----------------------------------------------------------------
# Taylor series coefficients of Jacobi SN, CN, DN
#----------------------------------------------------------------
n := 6: g := x: for i from 1 to 2*n do g := simplify(y*z*diff(g, x) + x*z*diff(g, y) + x*y*diff(g, z)); if(type(i, odd))then SN := simplify(sort(subs(z = k, subs(y = 1, subs(x = 0, g)))) / k);
# lprint("SN ", SN); lprint("SN ", seq(coeff(SN, k, j), j=0..i-1, 2)); else CN := simplify(sort(subs(z = 1, subs(y = 0, subs(x = k, g)))) / k); DN := simplify(sort(subs(z = 0, subs(y = k, subs(x = 1, g)))));
# lprint("CN ", CN); # lprint("DN ", DN); lprint("CN ", seq(coeff(CN, k, j), j=0..i-2, 2)); lprint("DN ", seq(coeff(DN, k, j), j=2..i, 2)); end; end: (End)
A060628 := proc(n, m) JacobiSN(z, k) ; coeftayl(%, z=0, 2*n+1) ; (-1)^n*coeftayl(%, k=0, 2*m)*(2*n+1)! ; end proc: # alternative program, R. J. Mathar, Jan 30 2011
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MATHEMATICA
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maxn = 8; se = Series[ JacobiSN[u, m], {u, 0, 2*maxn + 1 }]; cc = Partition[ CoefficientList[se, u], 2][[All, 2]]; Flatten[ (CoefficientList[#, m] & /@ cc)* Table[(-1)^n*(2*n + 1)!, {n, 0, maxn}]] (* Jean-François Alcover, Sep 21 2011 *)
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CROSSREFS
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Row sums give A000182. Diagonals: A004004, A004005, A002753, A032348, A032427, A032428, A032429, A032430, A032431, A032432, A032433.
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KEYWORD
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AUTHOR
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STATUS
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approved
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