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%I #48 Feb 24 2023 11:18:38
%S 1,0,1,0,2,1,0,3,4,1,0,4,10,6,1,0,5,20,21,8,1,0,6,35,56,36,10,1,0,7,
%T 56,126,120,55,12,1,0,8,84,252,330,220,78,14,1,0,9,120,462,792,715,
%U 364,105,16,1,0,10,165,792,1716,2002,1365,560,136,18,1
%N Riordan array (1, x/(1-x)^2).
%C Triangle T(n,k), 0 <= k <= n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C Row sums give A088305. - _Philippe Deléham_, Nov 21 2007
%C Column k is C(n,2k-1) for k > 0. - _Philippe Deléham_, Jan 20 2012
%C From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - _Tom Copeland_, Oct 11 2014
%C T is the convolution triangle of the positive integers (see A357368). - _Peter Luschny_, Oct 19 2022
%H G. C. Greubel, <a href="/A128908/b128908.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv:1110.6620 [math.RT], 2014.
%F T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
%F Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - _Philippe Deléham_, Oct 19 2008
%F G.f.: (1-x)^2/(1-(2+y)*x+x^2). - _Philippe Deléham_, Jan 20 2012
%F Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A001352(n), (-1)^(n+1)*A054888(n+1), (-1)^n*A008574(n), (-1)^n*A084103(n), (-1)^n*A084099(n), A163810(n), A000007(n), A088305(n) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively. - _Philippe Deléham_, Jan 20 2012
%F Riordan array (1, x/(1-x)^2). - _Philippe Deléham_, Jan 20 2012
%e The triangle T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10
%e 0: 1
%e 1: 0 1
%e 2: 0 2 1
%e 3: 0 3 4 1
%e 4: 0 4 10 6 1
%e 5: 0 5 20 21 8 1
%e 6: 0 6 35 56 36 10 1
%e 7: 0 7 56 126 120 55 12 1
%e 8: 0 8 84 252 330 220 78 14 1
%e 9: 0 9 120 462 792 715 364 105 16 1
%e 10: 0 10 165 792 1716 2002 1365 560 136 18 1
%e ... reformatted by _Wolfdieter Lang_, Jul 31 2017
%e From _Peter Luschny_, Mar 06 2022: (Start)
%e The sequence can also be seen as a square array read by upwards antidiagonals.
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%e 0, 2, 4, 6, 8, 10, 12, 14, 16, ... A005843
%e 0, 3, 10, 21, 36, 55, 78, 105, 136, ... A014105
%e 0, 4, 20, 56, 120, 220, 364, 560, 816, ... A002492
%e 0, 5, 35, 126, 330, 715, 1365, 2380, 3876, ... (A053126)
%e 0, 6, 56, 252, 792, 2002, 4368, 8568, 15504, ... (A053127)
%e 0, 7, 84, 462, 1716, 5005, 12376, 27132, 54264, ... (A053128)
%e 0, 8, 120, 792, 3432, 11440, 31824, 77520, 170544, ... (A053129)
%e 0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
%e A27,A292, A389, A580, A582, A1288, A10966, A10968, A165817 (End)
%p # Computing the rows of the array representation:
%p S := proc(n,k) option remember;
%p if n = k then 1 elif k < 0 or k > n then 0 else
%p S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
%p Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
%p for n from 0 to 8 do Arow(n, 9) od; # _Peter Luschny_, Mar 06 2022
%p # Uses function PMatrix from A357368.
%p PMatrix(10, n -> n); # _Peter Luschny_, Oct 19 2022
%t With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - x)^2/(1 - (2 + y)*x + x^2), {x, 0, nmax}, {y, 0, nmax}], x], y]] // Flatten (* _G. C. Greubel_, Nov 22 2017 *)
%o (Sage)
%o @cached_function
%o def T(k,n):
%o if k==n: return 1
%o if k==0: return 0
%o return sum(i*T(k-1,n-i) for i in (1..n-k+1))
%o A128908 = lambda n,k: T(k,n)
%o for n in (0..10): print([A128908(n,k) for k in (0..n)]) # _Peter Luschny_, Mar 12 2016
%o (PARI) for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, if(k==0, 0, binomial(n+k-1,2*k-1))), ", "))) \\ _G. C. Greubel_, Nov 22 2017
%o (Python)
%o from functools import cache
%o @cache
%o def A128908(n, k):
%o if n == k: return 1
%o if (k <= 0 or k > n): return 0
%o return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
%o for n in range(10):
%o print([A128908(n, k) for k in range(n+1)]) # _Peter Luschny_, Mar 07 2022
%Y Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
%Y Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
%Y Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
%Y Cf. A165817 (the main diagonal of the array).
%K nonn,tabl
%O 0,5
%A _Philippe Deléham_, Apr 22 2007
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