OFFSET
0,2
COMMENTS
Apart from signs, identical toA078812.
Another version with row-leading 0's and differing signs is given byA285072.
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. (seeA053121for 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 areA000027,A000292,A000389,A000580,A000582,A001288for m=0..5, resp. For m=6..23 they areA010966..(+2)..A011000and for m=24..49 they areA017713..(+2)..A017763.
Riordan array (1/(1+x)^2,x/(1+x)^2). Inverse array isA039598.Diagonal sums have g.f. 1/(1+x^2). -Paul Barry,Mar 17 2005. Corrected by _Wolfdieter_ Lang, Nov 13 2012.
Unsigned version is inA078812.-Philippe Deléham,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,May 23 2007
FromWolfdieter Lang,Nov 13 2012: (Start)
The A-sequence for this Riordan triangle isA115141,and the Z-sequence isA115141(n+1), n>=0. For A- and Z-sequences for Riordan matrices see the W. Lang link underA006232with details and references.
S(n,x^2-2) = sum(r(j,x^2),j=0..n) with Chebyshev's S-polynomials and r(j,x^2):= R(2*j+1,x)/x, where R(n,x) are the monic integer Chebyshv T-polynomials with coefficients given inA127672.Proof from comparing the o.g.f. of the partial sum of the r(j,x^2) polynomials (see a comment on the signed Riordan triangleA111125) with the present Riordan type o.g.f. for the row polynomials with x -> x^2. (End)
S(n,x^2-2) = S(2*n+1,x)/x, n >= 0, from the odd part of the bisection of the o.g.f. -Wolfdieter Lang,Dec 17 2012
For a relation to a generator for the Narayana numbersA001263,seeA119900,whose columns are unsigned shifted rows (or antidiagonals) of this array, referring to the tables in the example sections. -Tom Copeland,Oct 29 2014
The unsigned rows of this array are alternating rows of a mirroredA011973and alternating shifted rows ofA030528for the Fibonacci polynomials. -Tom Copeland,Nov 04 2014
Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment inA046521with references): a(n, m) = (2*(m + 1)/(n - m))*Sum_{k = m..n-1} (-1)^(n-k)*a(k, m), with input a(n, n) = 1, and a(n,k) = 0 for n < k. -Wolfdieter Lang,Jun 03 2020
Row n gives the characteristic polynomial of the (n X n)-matrix M where M[i,j] = 2 if i = j, -1 if |i-j| = 1 and 0 otherwise. The matrix M is positive definite and has 2-condition number (cot(Pi/(2*n+2)))^2. -Jianing Song,Jun 21 2022
Also the convolution triangle of (-1)^(n+1)*n. -Peter Luschny,Oct 07 2022
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.
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, Graduate 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
Wolfdieter Lang,First rows of the triangle.
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].
P. Barry,Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays,JIS 12 (2009) 09.8.6.
Naiomi T. Cameron and Asamoah Nkwanta,On some (pseudo) involutions in the Riordan group,J. of Integer Sequences, 8(2005), 1-16.
P. Damianou,On the characteristic polynomials of Cartan matrices and Chebyshev polynomials,arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). -Tom Copeland,Oct 11 2014
Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen,Some polynomials associated with regular polygons,Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
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;
O.g.f. for m-th column (signed triangle): ((x/(1+x)^2)^m)/(1+x)^2.
FromJianing Song,Jun 21 2022: (Start)
T(n,k) = [x^k]f_n(x), where f_{-1}(x) = 0, f_0(x) = 1, f_n(x) = (x-2)*f_{n-1}(x) - f_{n-2}(x) for n >= 2.
f_n(x) = (((x-2+sqrt(x^2-4*x))/2)^(n+1) - ((x-2-sqrt(x^2-4*x))/2)^(n+1))/sqrt(x^2-4x).
The roots of f_n(x) are 2 + 2*cos(k*Pi/(n+1)) = 4*(cos(k*Pi/(2*n+2)))^2 for 1 <= k <= n. (End)
EXAMPLE
The triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9
0: 1
1: -2 1
2: 3 -4 1
3: -4 10 -6 1
4: 5 -20 21 -8 1
5: -6 35 -56 36 -10 1
6: 7 -56 126 -120 55 -12 1
7: -8 84 -252 330 -220 78 -14 1
8: 9 -120 462 -792 715 -364 105 -16 1
9: -10 165 -792 1716 -2002 1365 -560 136 -18 1
... Reformatted and extended byWolfdieter Lang,Nov 13 2012
E.g., fourth row (n=3) {-4,10,-6,1} corresponds to the polynomial S(3,x-2) = -4+10*x-6*x^2+x^3.
FromWolfdieter Lang,Nov 13 2012: (Start)
Recurrence: a(5,1) = 35 = 1*5 + (-2)*(-20) -1*(10).
Recurrence from Z-sequence [-2,-1,-2,-5,...]: a(5,0) = -6 = (-2)*5 + (-1)*(-20) + (-2)*21 + (-5)*(-8) + (-14)*1.
Recurrence from A-sequence [1,-2,-1,-2,-5,...]: a(5,1) = 35 = 1*5 + (-2)*(-20) + (-1)*21 + (-2)*(-8) + (-5)*1.
(End)
E.g., the fourth row (n=3) {-4,10,-6,1} corresponds also to the polynomial S(7,x)/x = -4 + 10*x^2 - 6*x^4 + x^6. -Wolfdieter Lang,Dec 17 2012
Boas-Buck type recurrence: -56 = a(5, 2) = 2*(-1*1 + 1*(-6) - 1*21) = -2*28 = -56. -Wolfdieter Lang,Jun 03 2020
MAPLE
seq(seq((-1)^(n+m)*binomial(n+m+1, 2*m+1), m=0..n), n=0..10); #Robert Israel,Oct 15 2014
# Uses function PMatrix fromA357368.Adds a row above and a column to the left.
PMatrix(10, n -> -(-1)^n*n); #Peter Luschny,Oct 07 2022
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,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,May 23 2007 *)
PROG
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved