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%I #80 Jun 19 2020 04:41:38
%S 1,-1,1,1,-3,1,-1,6,-5,1,1,-10,15,-7,1,-1,15,-35,28,-9,1,1,-21,70,-84,
%T 45,-11,1,-1,28,-126,210,-165,66,-13,1,1,-36,210,-462,495,-286,91,-15,
%U 1,-1,45,-330,924,-1287,1001,-455,120,-17,1,1,-55,495,-1716,3003,-3003,1820,-680,153,-19,1
%N Riordan array (1/(1+x), x/(1+x)^2), inverse array is A039599.
%C This sequence is up to sign the same as A129818. - _T. D. Noe_, Sep 30 2011
%C Row sums: A057078. - _Philippe Deléham_, Jun 11 2007
%C Subtriangle of the triangle given by (0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 19 2012
%C This triangle provides the coefficients of powers of x^2 for the even-indexed Chebyshev S polynomials (see A049310): S(2*n,x) = Sum_{k=0..n} T(n,k)*x^(2*k), n >= 0. - _Wolfdieter Lang_, Dec 17 2012
%C If L(x^n) := C(n) = A000108(n) (Catalan numbers), then the polynomials P_n(x) := Sum_{k=0..n} T(n,k)*x^k are orthogonal with respect to the inner product given by (f(x),g(x)) := L(f(x)*g(x)). - _Michael Somos_, Jan 03 2019
%H Vincenzo Librandi, <a href="/A129818/b129818.txt">Rows n = 1..101, flattened</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H P. Barry, A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry2/barry94r.html">The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences</a>, J. Int. Seq. 13 (2010) # 10.8.2, example 15.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">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.
%F T(n,k) = (-1)^(n-k)*A085478(n,k) = (-1)^(n-k)*binomial(n+k,2*k).
%F Sum_{k=0..n} T(n,k)*A000531(k) = n^2, with A000531(0)=0. - _Philippe Deléham_, Jun 11 2007
%F Sum_{k=0..n} T(n,k)*x^k = A033999(n), A057078(n), A057077(n), A057079(n), A005408(n), A002878(n), A001834(n), A030221(n), A002315(n), A033890(n), A057080(n), A057081(n), A054320(n), A097783(n), A077416(n), A126866(n), A028230(n+1) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16, respectively. - _Philippe Deléham_, Nov 19 2009
%F O.g.f.: (1+x)/(1+(2-y)*x+x^2). - _Wolfdieter Lang_, Dec 15 2010
%F O.g.f. column k with leading zeros (Riordan array, see NAME): (1/(1+x))*(x/(1+x)^2)^k, k >= 0. - _Wolfdieter Lang_, Dec 15 2010
%F From _Wolfdieter Lang_, Dec 20 2010: (Start)
%F Recurrences from the Z- and A-sequences for Riordan arrays. See the W. Lang link under A006232 for details and references.
%F T(n,0) = -1*T(n-1,0), n >= 1, from the o.g.f. -1 for the Z-sequence (trivial result).
%F T(n,k) = Sum_{j=0..n-k} A(j)*T(n-1,k-1+j), n >= k >= 1, with A(j):= A115141(j) = [1,-2,-1,-2,-5,-14,...], j >= 0 (o.g.f. 1/c(x)^2 with the A000108 (Catalan) o.g.f. c(x)). (End)
%F T(n,k) = (-1)^n*A123970(n,k). - _Philippe Deléham_, Feb 18 2012
%F T(n,k) = -2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -1, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 19 2012
%F A039599(m,n) = Sum_{k=0..n} T(n,k) * C(k+m) where C(n) are the Catalan numbers. - _Michael Somos_, Jan 03 2019
%F Equals the matrix inverse of the Riordan square (cf. A321620) of the Catalan numbers. - _Peter Luschny_, Jan 04 2019
%F Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): T(n,k) = (1 + 2*k)/(n - k))*Sum_{j = k..n-1} (-1)^(n-j)*T(j,m), with input T(n,n) = 1, and T(n,k) = 0 for n < k. - _Wolfdieter Lang_, Jun 03 2020
%e Triangle T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: -1 1
%e 2: 1 -3 1
%e 3: -1 6 -5 1
%e 4: 1 -10 15 -7 1
%e 5: -1 15 -35 28 -9 1
%e 6: 1 -21 70 -84 45 -11 1
%e 7: -1 28 -126 210 -165 66 -13 1
%e 8: 1 -36 210 -462 495 -286 91 -15 1
%e 9: -1 45 -330 924 -1287 1001 -455 120 -17 1
%e 10: 1 -55 495 -1716 3003 -3003 1820 -680 153 -19 1
%e ... Reformatted by _Wolfdieter Lang_, Dec 17 2012
%e Recurrence from the A-sequence A115141:
%e 15 = T(4,2) = 1*6 + (-2)*(-5) + (-1)*1.
%e (0, -1, 0, -1, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, ...) begins:
%e 1
%e 0, 1
%e 0, -1, 1
%e 0, 1, -3, 1
%e 0, -1, 6, -5, 1
%e 0, 1, -10, 15, -7, 1
%e 0, -1, 15, -35, 28, -9, 1. - _Philippe Deléham_, Mar 19 2012
%e Row polynomial for n=3 in terms of x^2: S(6,x) = -1 + 6*x^2 -5*x^4 + 1*x^6, with Chebyshev's S polynomial. See a comment above. - _Wolfdieter Lang_, Dec 17 2012
%e Boas-Buck type recurrence: -35 = T(5,2) = (5/3)*(-1*1 +1*(-5) - 1*15) = -3*7 = -35. - _Wolfdieter Lang_, Jun 03 2020
%p # The function RiordanSquare is defined in A321620.
%p RiordanSquare((1 - sqrt(1 - 4*x))/(2*x), 10):
%p LinearAlgebra[MatrixInverse](%); # _Peter Luschny_, Jan 04 2019
%t max = 10; Flatten[ CoefficientList[#, y] & /@ CoefficientList[ Series[ (1 + x)/(1 + (2 - y)*x + x^2), {x, 0, max}], x]] (* _Jean-François Alcover_, Sep 29 2011, after _Wolfdieter Lang_ *)
%o (Sage)
%o @CachedFunction
%o def A129818(n,k):
%o if n< 0: return 0
%o if n==0: return 1 if k == 0 else 0
%o h = A129818(n-1,k) if n==1 else 2*A129818(n-1,k)
%o return A129818(n-1,k-1) - A129818(n-2,k) - h
%o for n in (0..9): [A129818(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y Cf. A039599, A085478, A123970, A321620.
%K sign,tabl
%O 0,5
%A _Philippe Deléham_, Jun 09 2007
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