This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A128908 #48 Feb 24 2023 11:18:38
%S A128908 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 A128908 56,126,120,55,12,1,0,8,84,252,330,220,78,14,1,0,9,120,462,792,715,
%U A128908 364,105,16,1,0,10,165,792,1716,2002,1365,560,136,18,1
%N A128908 Riordan array (1, x/(1-x)^2).
%C A128908 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 A128908 Row sums give A088305. - _Philippe Deléham_, Nov 21 2007
%C A128908 Column k is C(n,2k-1) for k > 0. - _Philippe Deléham_, Jan 20 2012
%C A128908 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 A128908 T is the convolution triangle of the positive integers (see A357368). - _Peter Luschny_, Oct 19 2022
%H A128908 G. C. Greubel, <a href="/A128908/b128908.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H A128908 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 A128908 T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
%F A128908 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 A128908 G.f.: (1-x)^2/(1-(2+y)*x+x^2). - _Philippe Deléham_, Jan 20 2012
%F A128908 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 A128908 Riordan array (1, x/(1-x)^2). - _Philippe Deléham_, Jan 20 2012
%e A128908 The triangle T(n,k) begins:
%e A128908 n\k 0 1 2 3 4 5 6 7 8 9 10
%e A128908 0: 1
%e A128908 1: 0 1
%e A128908 2: 0 2 1
%e A128908 3: 0 3 4 1
%e A128908 4: 0 4 10 6 1
%e A128908 5: 0 5 20 21 8 1
%e A128908 6: 0 6 35 56 36 10 1
%e A128908 7: 0 7 56 126 120 55 12 1
%e A128908 8: 0 8 84 252 330 220 78 14 1
%e A128908 9: 0 9 120 462 792 715 364 105 16 1
%e A128908 10: 0 10 165 792 1716 2002 1365 560 136 18 1
%e A128908 ... reformatted by _Wolfdieter Lang_, Jul 31 2017
%e A128908 From _Peter Luschny_, Mar 06 2022: (Start)
%e A128908 The sequence can also be seen as a square array read by upwards antidiagonals.
%e A128908 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%e A128908 0, 2, 4, 6, 8, 10, 12, 14, 16, ... A005843
%e A128908 0, 3, 10, 21, 36, 55, 78, 105, 136, ... A014105
%e A128908 0, 4, 20, 56, 120, 220, 364, 560, 816, ... A002492
%e A128908 0, 5, 35, 126, 330, 715, 1365, 2380, 3876, ... (A053126)
%e A128908 0, 6, 56, 252, 792, 2002, 4368, 8568, 15504, ... (A053127)
%e A128908 0, 7, 84, 462, 1716, 5005, 12376, 27132, 54264, ... (A053128)
%e A128908 0, 8, 120, 792, 3432, 11440, 31824, 77520, 170544, ... (A053129)
%e A128908 0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
%e A128908 A27,A292, A389, A580, A582, A1288, A10966, A10968, A165817 (End)
%p A128908 # Computing the rows of the array representation:
%p A128908 S := proc(n,k) option remember;
%p A128908 if n = k then 1 elif k < 0 or k > n then 0 else
%p A128908 S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
%p A128908 Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
%p A128908 for n from 0 to 8 do Arow(n, 9) od; # _Peter Luschny_, Mar 06 2022
%p A128908 # Uses function PMatrix from A357368.
%p A128908 PMatrix(10, n -> n); # _Peter Luschny_, Oct 19 2022
%t A128908 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 A128908 (Sage)
%o A128908 @cached_function
%o A128908 def T(k,n):
%o A128908 if k==n: return 1
%o A128908 if k==0: return 0
%o A128908 return sum(i*T(k-1,n-i) for i in (1..n-k+1))
%o A128908 A128908 = lambda n,k: T(k,n)
%o A128908 for n in (0..10): print([A128908(n,k) for k in (0..n)]) # _Peter Luschny_, Mar 12 2016
%o A128908 (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 A128908 (Python)
%o A128908 from functools import cache
%o A128908 @cache
%o A128908 def A128908(n, k):
%o A128908 if n == k: return 1
%o A128908 if (k <= 0 or k > n): return 0
%o A128908 return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
%o A128908 for n in range(10):
%o A128908 print([A128908(n, k) for k in range(n+1)]) # _Peter Luschny_, Mar 07 2022
%Y A128908 Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
%Y A128908 Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
%Y A128908 Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
%Y A128908 Cf. A165817 (the main diagonal of the array).
%K A128908 nonn,tabl
%O A128908 0,5
%A A128908 _Philippe Deléham_, Apr 22 2007