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 A128908 Riordan array (1, x/(1-x)^2). 9
 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, 56, 126, 120, 55, 12, 1, 0, 8, 84, 252, 330, 220, 78, 14, 1, 0, 9, 120, 462, 792, 715, 364, 105, 16, 1, 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. Row sums give A088305. - Philippe Deléham, Nov 21 2007 Column k is C(n,2k-1) for k > 0. - Philippe Deléham, Jan 20 2012 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 LINKS G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014. FORMULA T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1. Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - Philippe Deléham, Oct 19 2008 G.f.: (1-x)^2/(1-(2+y)*x+x^2). - Philippe Deléham, Jan 20 2012 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 Riordan array (1, x/(1-x)^2). - Philippe Deléham, Jan 20 2012 EXAMPLE The triangle  T(n,k) begins:    n\k  0    1    2    3    4    5    6    7    8    9   10    0:   1    1:   0    1    2:   0    2    1    3:   0    3    4    1    4:   0    4   10    6    1    5:   0    5   20   21    8    1    6:   0    6   35   56   36   10    1    7:   0    7   56  126  120   55   12    1    8:   0    8   84  252  330  220   78   14    1    9:   0    9  120  462  792  715  364  105   16    1   10:   0   10  165  792 1716 2002 1365  560  136   18    1   ... reformatted by Wolfdieter Lang, Jul 31 2017 MATHEMATICA 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 *) PROG (Sage) @cached_function def T(k, n):     if k==n: return 1     if k==0: return 0     return sum(i*T(k-1, n-i) for i in (1..n-k+1)) A128908 = lambda n, k: T(k, n) for n in (0..10): print [A128908(n, k) for k in (0..n)] # Peter Luschny, Mar 12 2016 (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 CROSSREFS Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305. Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763. Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810. Sequence in context: A220399 A268830 A095884 * A285072 A300454 A155112 Adjacent sequences:  A128905 A128906 A128907 * A128909 A128910 A128911 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Apr 22 2007 STATUS approved

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Last modified August 24 16:16 EDT 2019. Contains 326295 sequences. (Running on oeis4.)