%I #61 Jun 03 2024 14:26:47

%S 1,0,1,-1,0,1,-1,-2,0,1,0,-2,-3,0,1,1,1,-3,-4,0,1,1,4,3,-4,-5,0,1,0,3,

%T 9,6,-5,-6,0,1,-1,-2,5,16,10,-6,-7,0,1,-1,-6,-9,6,25,15,-7,-8,0,1,0,

%U -4,-18,-24,5,36,21,-8,-9,0,1,1,3,-7,-39,-50,1,49,28,-9,-10,0,1,1,8

%N Triangle T read by rows: inverse of Motzkin triangle A097609.

%C Riordan array ((1-x)/(1-x+x^2),x(1-x)/(1-x+x^2)). - _Paul Barry_, Jun 21 2008

%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="https://doi.org/10.1007/978-3-0348-8405-1_11">An algebra for proper generating trees</a>, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [<a href="https://www.researchgate.net/publication/2386023_An_Algebra_for_Proper_Generating_Trees">alternative link</a>]

%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://local.disia.unifi.it/merlini/papers/Lucidi.ps">An algebra for proper generating trees</a>, Colloquium on Mathematics and Computer Science, Versailles, September 2000.

%F T(n,m) = sum(j=0..m, binomial(m,j)*sum(k=0..n, binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1))*(-1)^(m-j)). - _Vladimir Kruchinin_, Apr 08 2011

%F T(n,m) = sum(k=ceiling((n-m-1)/2)..n-m, binomial(k+m,m)*binomial(k+1,n-k-m)*(-1)^(n-k-m)). - _Vladimir Kruchinin_, Dec 17 2011

%F T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Feb 20 2013

%F T(n+5,n) = (n+1)^2. - _Philippe Deléham_, Feb 20 2013

%F From _Tom Copeland_, Nov 01 and 04 2014: (Start)

%F O.g.f.: G(x,t) = Pinv[Cinv(x),t+1] = Cinv(x) / [1 - (t+1)Cinv(x)] = x*(1-x) / [1-(t+1)x(1-x)] = x + t * x^2 + (-1 + t^2) * x^3 + ..., where Cinv(x)= x * (1-x) is the inverse of C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the Catalan numbers A000108 and Pinv(x,t) = -P(-x,t) = x/(1-t*x) is the inverse of P(x,t) = x/(1+x*t).

%F Ginv(x,t)= C[P[x,t+1]]= C[x/(1+(t+1)x)] = {1-sqrt[1-4*x/(1+(t+1)x)]}/2.

%F The inverse in x of G(x,t) with t replaced by -t is the o.g.f. of A091867, and G(x,t-1) is a signed version of the (mirrored) Fibonacci polynomials A030528. (End)

%e 1

%e 0,1

%e -1,0,1

%e -1,-2,0,1

%e 0,-2,-3,0,1

%e 1,1,-3,-4,0,1

%e 1,4,3,-4,-5,0,1

%e 0,3,9,6,-5,-6,0,1

%e -1,-2,5,16,10,-6,-7,0,1

%e -1,-6,-9,6,25,15,-7,-8,0,1

%p # Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.

%p InvPMatrix(10, n -> A005043(n-1)); # _Peter Luschny_, Oct 09 2022

%o (Maxima)

%o T(n,m):=sum(binomial(m,j)*sum(binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1),k,0,n)*(-1)^(m-j),j,0,m); /* _Vladimir Kruchinin_, Apr 08 2011 */

%Y Row sums are A009116 with different signs.

%Y Row sums are A146559(n).

%Y Cf. A005043, A091867, A030528, A000108.

%K sign,tabl

%O 0,8

%A _Ralf Stephan_, Mar 17 2005