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A128908
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Riordan array (1, x/(1-x)^2).
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9
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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: A268830 A095884 A342240 * A285072 A300454 A155112
Adjacent sequences: A128905 A128906 A128907 * A128909 A128910 A128911
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe Deléham, Apr 22 2007
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STATUS
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approved
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