|
| |
|
|
A053122
|
|
Triangle of coefficients of Chebyshev's S(n,x-2)= U(n,x/2-1) polynomials (exponents of x in increasing order).
|
|
15
| |
|
|
1, -2, 1, 3, -4, 1, -4, 10, -6, 1, 5, -20, 21, -8, 1, -6, 35, -56, 36, -10, 1, 7, -56, 126, -120, 55, -12, 1, -8, 84, -252, 330, -220, 78, -14, 1, 9, -120, 462, -792, 715, -364, 105, -16, 1, -10, 165, -792, 1716, -2002, 1365, -560, 136, -18, 1, 11, -220, 1287, -3432, 5005, -4368, 2380, -816, 171, -20
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Apart from signs, identical to A078812.
G.f. for row polynomials S(n,x-2) (signed triangle): 1/(1+(2-x)*z+z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-(2+x)*z+z^2) for row polynomials.
Row sums (signed triangle) A049347(n) (periodic(1,-1,0)). Row sums (unsigned triangle) A001906(n+1)=F(2*(n+1)) (even indexed Fibonacci).
In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The (unsigned) column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for m=0..5, resp. For m=6..23 they are A010966..(+2)..A011000 and for m=24..49 they are A017713..(+2)..A017763.
Riordan array (1/(1+x)^2,x/(1+x^2)). Inverse array is A039598. Diagonal sums have g.f. 1/(1+x^2). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2005
Unsigned version is in A078812 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
Also row n gives (except for an overall sign) coefficients of characteristic polynomial of the Cartan matrix for the root system A_n. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 23 2007
|
|
|
REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62
Sigurdur Helgasson,Differential Geometry, Lie Groups and Symmetric Spaces,Graduaste Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978,p. 463
|
|
|
LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Chebyshev polynomials.
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram
|
|
|
FORMULA
| a(n, m) := 0 if n<m else ((-1)^(n-m))*binomial(n+m+1, 2*m+1);
a(n, m) = -2*a(n-1, m) + a(n-1, m-1) - a(n-2, m), a(n, -1) := 0 =: a(-1, m), a(0, 0)=1, a(n, m) := 0 if n<m; G.f. for m-th column (signed triangle):((x/(1+x)^2)^m)/(1+x)^2.
|
|
|
EXAMPLE
| {1}; {-2,1}; {3,-4,1}; {-4,10,-6,1}; {5,-20,21,-8};... E.g. fourth row (n=3) {-4,10,-6,1} corresponds to polynomial S(3,x-2)= -4+10*x-6*x^2+x^3.
|
|
|
MATHEMATICA
| T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 23 2007
Alternative code for the matrices from MathWorld: sln[n_] := 2IdentityMatrix[n] - PadLeft[PadRight[IdentityMatrix[n - 1], {n, n - 1}], {n, n}] - PadLeft[PadRight[IdentityMatrix[n - 1], {n - 1, n}], {n, n}] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 23 2007
|
|
|
CROSSREFS
| Cf. A005248, A127677, A053123, A049310.
Sequence in context: A143326 A186686 * A078812 A104711 A133112 A198060
Adjacent sequences: A053119 A053120 A053121 * A053123 A053124 A053125
|
|
|
KEYWORD
| easy,nice,sign,tabl
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
|
| |
|
|
