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 A104597 Triangle T read by rows: inverse of Motzkin triangle A097609. 13
 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, 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, -4, -18, -24, 5, 36, 21, -8, -9, 0, 1, 1, 3, -7, -39, -50, 1, 49, 28, -9, -10, 0, 1, 1, 8 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Riordan array ((1-x)/(1-x+x^2),x(1-x)/(1-x+x^2)). - Paul Barry, Jun 21 2008 LINKS D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link] D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000. FORMULA 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 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 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 T(n+5,n) = (n+1)^2. - Philippe Deléham, Feb 20 2013 From Tom Copeland, Nov 01 and 04 2014: (Start) 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). Ginv(x,t)= C[P[x,t+1]]= C[x/(1+(t+1)x)] = {1-sqrt[1-4*x/(1+(t+1)x)]}/2. 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) EXAMPLE 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 0,3,9,6,-5,-6,0,1 -1,-2,5,16,10,-6,-7,0,1 -1,-6,-9,6,25,15,-7,-8,0,1 PROG (Maxima) 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 CROSSREFS Row sums are A009116 with different signs. Row sums are A146559(n). - Philippe Deléham, Feb 20 2013 Cf. A091867, A030528, A000108. Sequence in context: A125079 A329027 A235987 * A182936 A340503 A072662 Adjacent sequences:  A104594 A104595 A104596 * A104598 A104599 A104600 KEYWORD sign,tabl AUTHOR Ralf Stephan, Mar 17 2005 STATUS approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)