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A104562 Inverse of the Motzkin triangle A064189.

%I A104562 #71 Sep 01 2024 18:27:29
%S A104562 1,-1,1,0,-2,1,1,1,-3,1,-1,2,3,-4,1,0,-4,2,6,-5,1,1,2,-9,0,10,-6,1,-1,
%T A104562 3,9,-15,-5,15,-7,1,0,-6,3,24,-20,-14,21,-8,1,1,3,-18,-6,49,-21,-28,
%U A104562 28,-9,1,-1,4,18,-36,-35,84,-14,-48,36,-10,1,0,-8,4,60,-50,-98,126,6,-75,45,-11,1,1,4,-30,-20,145,-36,-210,168,45,-110,55,-12,1
%N A104562 Inverse of the Motzkin triangle A064189.
%C A104562 Or, triangle read by rows: T(0, 0) = 1; for n >= 1 T(n, k) is the coefficient of x^k in the monic characteristic polynomial of the n X n tridiagonal matrix with 1's on the main, sub- and superdiagonal (0 <= k <= n). The characteristic polynomial has a root 1 + 2*cos(Pi/(n + 1)). - _Gary W. Adamson_, Nov 19 2006
%C A104562 Row sums have g.f. 1/(1 + x^2); diagonal sums are (-1)^n. Riordan array (1/(1 + x + x^2), x/(1 + x + x^2)).
%C A104562 Or, triangle read by rows in which row n gives coefficients of characteristic polynomial of the n X n tridiagonal matrix with 1's on the main diagonal and -1's on the two adjacent diagonals. For example: M(3) = {{1, -1, 0}, {-1, 1, -1}, {0, -1, 1}}. - _Roger L. Bagula_, Mar 15 2008
%C A104562 Subtriangle of the triangle given by [0,-1,1,-1,0,0,0,0,0,0,0,...) DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 27 2010
%C A104562 Triangle of coefficients of Chebyshev's S(n, x-1) polynomials (exponents of x in increasing order). - _Philippe Deléham_, Feb 19 2012
%D A104562 Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
%H A104562 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry4/bern2.html">Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences</a>, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.
%H A104562 Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, and David G. L. Wang, <a href="https://doi.org/10.1016/j.jmaa.2016.04.033">Root geometry of polynomial sequences. II: Type (1,0)</a>, J. Math. Anal. Appl.  441, No. 2, 499-528 (2016).
%H A104562 A. Luzón, D. Merlini, M. A. Morón, and R. Sprugnoli, <a href="http://dx.doi.org/10.1016/j.dam.2014.03.005">Complementary Riordan arrays</a>, Discrete Applied Mathematics, 172 (2014) 75-87.
%F A104562 T(n, k) = Sum_{j=0..n} (-1)^(k-j)*(-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2.
%F A104562 T(n,k) = (-1)^(n-k)*A101950(n,k). - _Philippe Deléham_, Feb 19 2012
%F A104562 T(n,k) = T(n-1,k-1) - T(n-1,k) - T(n-2,l). - _Philippe Deléham_, Feb 19 2012
%F A104562 A104562*A007318 = A049310 as infinite lower triangular matrices. - _Philippe Deléham_, Feb 19 2012
%F A104562 G.f.: 1/(1+x+x^2-y*x). - _Philippe Deléham_, Feb 19 2012
%F A104562 T(n, k) = (-1)^(n - k)*C(n, k)*hypergeom([(k - n)/2, (k - n + 1)/2], [-n], 4) for n >= 1. - _Peter Luschny_, Apr 25 2016
%e A104562 Triangle starts:
%e A104562   [0]  1;
%e A104562   [1] -1,  1;
%e A104562   [2]  0, -2,   1;
%e A104562   [3]  1,  1,  -3,   1;
%e A104562   [4] -1,  2,   3,  -4,   1;
%e A104562   [5]  0, -4,   2,   6,  -5,   1;
%e A104562   [6]  1,  2,  -9,   0,  10,  -6,   1;
%e A104562   [7] -1,  3,   9, -15,  -5,  15,  -7,  1;
%e A104562   [8]  0, -6,   3,  24, -20, -14,  21, -8,  1;
%e A104562   [9]  1,  3, -18,  -6,  49, -21, -28, 28, -9, 1.
%e A104562   ...
%e A104562 From _Philippe Deléham_, Jan 27 2010: (Start)
%e A104562 Triangle [0,-1,1,-1,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] begins:
%e A104562   1;
%e A104562   0,  1;
%e A104562   0, -1,  1;
%e A104562   0,  0, -2,  1;
%e A104562   0,  1,  1, -3,  1;
%e A104562   0, -1,  2,  3, -4,  1;
%e A104562   ... (End)
%p A104562 with(linalg): m:=proc(i,j) if abs(i-j)<=1 then 1 else 0 fi end: T:=(n,k)->coeff(charpoly(matrix(n,n,m),x),x,k): 1; for n from 1 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%p A104562 # Alternatively:
%p A104562 T := (n,k) -> `if`(n=0,1,(-1)^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], 4)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # _Peter Luschny_, Apr 25 2016
%t A104562 nmax = 12;
%t A104562 M[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, k+2, 4];
%t A104562 invM = Inverse@Table[M[n, k], {n, 0, nmax}, {k, 0, nmax}];
%t A104562 T[n_, k_] := invM[[n+1, k+1]];
%t A104562 Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 15 2023 *)
%o A104562 (Sage)
%o A104562 @CachedFunction
%o A104562 def A104562(n,k):
%o A104562     if n< 0: return 0
%o A104562     if n==0: return 1 if k == 0 else 0
%o A104562     return A104562(n-1,k-1)-A104562(n-2,k)-A104562(n-1,k)
%o A104562 for n in (0..9): [A104562(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%o A104562 (Sage) # Alternatively as coefficients of polynomials:
%o A104562 def S(n,x):
%o A104562     if n==0: return 1
%o A104562     if n==1: return x-1
%o A104562     return (x-1)*S(n-1,x)-S(n-2,x)
%o A104562 for n in (0..7): print(S(n,x).list()) # _Peter Luschny_, Jun 23 2015
%Y A104562 Apart from signs identical to A101950.
%Y A104562 Cf. A125090.
%K A104562 easy,sign,tabl
%O A104562 0,5
%A A104562 _Paul Barry_, Mar 15 2005
%E A104562 Edited by _N. J. A. Sloane_, Apr 10 2008
%E A104562 Typo correction in the _Roger L. Bagula_ comment and Mathematica section by _Wolfdieter Lang_, Nov 22 2011