A108045 - OEIS

%I #39 Nov 24 2024 11:28:55

%S 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,

%T 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,

%U -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

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

%C Signed version of A114525. - _Eric W. Weisstein_, Apr 07 2017

%C For n >= 3, also the coefficients of the matching polynomial for the n-cycle graph C_n. - _Eric W. Weisstein_, Apr 07 2017

%C 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

%H G. C. Greubel, <a href="/A108045/b108045.txt">Rows n=0..100 of triangle, flattened</a>

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H P. Barry, A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.html">Meixner-Type Results for Riordan Arrays and Associated Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.9.4, section 5.

%H P. Peart and W.-J. Woan, <a href="http://dx.doi.org/10.1016/S0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

%H L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan Group</a>, Discrete Appl. Maths. 34 (1991) 229-239.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingPolynomial.html">Matching Polynomial</a>

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

%F G.f.: (1-x^2)/(1+x^2-tx). - _Emeric Deutsch_, Jun 06 2005

%F From _Peter Bala_, Jun 29 2015: (Start)

%F 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).

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

%F 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)

%e Triangle begins:

%e    1;

%e    0,  1;

%e   -2,  0,  1;

%e    0, -3,  0,  1;

%e    2,  0, -4,  0,  1;

%p 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

%t 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_ *)

%t Flatten[{{1}, CoefficientList[Table[I^n LucasL[n, -I x], {n, 10}], x]}] (* _Eric W. Weisstein_, Apr 07 2017 *)

%t Flatten[{{1}, CoefficientList[LinearRecurrence[{x, -1}, {x, -2 + x^2}, 10], x]}] (* _Eric W. Weisstein_, Apr 07 2017 *)

%Y Cf. A114525 (unsigned version).

%Y Cf. A007318, A108044.

%Y Cf. A127672.

%K sign,tabl,easy,changed

%O 0,4

%A _N. J. A. Sloane_, Jun 02 2005

%E More terms from _Emeric Deutsch_ and _Christian G. Bower_, Jun 06 2005