A108045 Triangle read by rows: lower triangular matrix obtained by inverting the lower triangular matrix in A108044.

1, 0, 1, -2, 0, 1, 0, -3, 0, 1, 2, 0, -4, 0, 1, 0, 5, 0, -5, 0, 1, -2, 0, 9, 0, -6, 0, 1, 0, -7, 0, 14, 0, -7, 0, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, -2, 0, 25, 0, -50, 0, 35, 0, -10, 0, 1, 0, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 2, 0, -36, 0, 105, 0, -112, 0, 54, 0, -12, 0, 1

Offset: 0

Author

N. J. A. Sloane, Jun 02 2005

Keywords

• sign
• tabl
• easy

Comments

Signed version of A114525. - Eric W. Weisstein, Apr 07 2017

For n >= 3, also the coefficients of the matching polynomial for the n-cycle graph C_n. - Eric W. Weisstein, Apr 07 2017

This triangle describes the Chebyshev transform of A100047 and following. Chebyshev transform of sequence b is c(n) = Sum_{k=1..n} a(n,k)*b(k). - Christian G. Bower, Jun 12 2005

Examples

			Triangle begins:
   1;
   0,  1;
  -2,  0,  1;
   0, -3,  0,  1;
   2,  0, -4,  0,  1;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
• Paul Barry and Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
• P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
• L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
• Eric Weisstein's World of Mathematics, Cycle Graph
• Eric Weisstein's World of Mathematics, Matching Polynomial

Crossrefs

Cf. A114525 (unsigned version).

Cf. A007318, A108044.

Cf. A127672.

Programs

Maple

f:=(1-x^2)/(1+x^2): g:=x/(1+x^2): G:=simplify(f/(1-t*g)): Gser:=simplify(series(G,x=0,14)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,x^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form # Emeric Deutsch, Jun 06 2005

Mathematica

a[n_, k_] := SeriesCoefficient[(1-x^2)/(1+x^2-t*x), {x, 0, n}, {t, 0, k}]; a[0, 0] = 1; Table[a[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2014, after Emeric Deutsch *)
Flatten[{{1}, CoefficientList[Table[I^n LucasL[n, -I x], {n, 10}], x]}] (* Eric W. Weisstein, Apr 07 2017 *)
Flatten[{{1}, CoefficientList[LinearRecurrence[{x, -1}, {x, -2 + x^2}, 10], x]}] (* Eric W. Weisstein, Apr 07 2017 *)

Formula

Riordan array ( (1-x^2)/(1+x^2), x/(1+x^2)).

G.f.: (1-x^2)/(1+x^2-tx). - Emeric Deutsch, Jun 06 2005

From Peter Bala, Jun 29 2015: (Start)

Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x/(1 + x^2) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).

T(n,k) = [x^(n-k)] f(x)^n with f(x) = ( 1 + sqrt(1 - 4*x^2) )/2.

In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

Extensions

More terms from Emeric Deutsch and Christian G. Bower, Jun 06 2005