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A099039
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Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.
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17
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1, 0, 1, 0, -1, 1, 0, 2, -2, 1, 0, -5, 5, -3, 1, 0, 14, -14, 9, -4, 1, 0, -42, 42, -28, 14, -5, 1, 0, 132, -132, 90, -48, 20, -6, 1, 0, -429, 429, -297, 165, -75, 27, -7, 1, 0, 1430, -1430, 1001, -572, 275, -110, 35, -8, 1, 0, -4862, 4862, -3432, 2002, -1001, 429, -154, 44, -9, 1, 0, 16796, -16796, 11934, -7072, 3640, -1638
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Row sums are generalized Catalan numbers A064310. Diagonal sums are 0^n+(-1)^n*A030238(n-2). Inverse is A026729, as number triangle. Columns have g.f. (xc(-x))^k=((sqrt(1+4x)-1)/2)^k.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, . . . ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, . . . ] where DELTA is the operator defined in A084938. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005
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REFERENCES
| F. R. Bernhart, Catalan, Mozkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
E. Deutsch, Dyck path enumeration, Discrete Math., 204 (1999), 167-202.
L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
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LINKS
| D. Callan, A recursive bijective approach to counting permutations . . .
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00. 1. 6
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations.
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FORMULA
| T(n, k) = (-1)^(n+k)*binomial(2n-k-1, n-k)*k/n for 0<=k<=n with n>0; T(0, 0) = 1; T(0, k) = 0 if k>0 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005
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EXAMPLE
| Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1},...
Triangle begins
1,
0, 1,
0, -1, 1,
0, 2, -2, 1,
0, -5, 5, -3, 1,
0, 14, -14, 9, -4, 1,
0, -42, 42, -28, 14, -5, 1,
0, 132, -132, 90, -48, 20, -6, 1,
0, -429, 429, -297, 165, -75, 27, -7, 1
Production matrix is
0, 1,
0, -1, 1,
0, 1, -1, 1,
0, -1, 1, -1, 1,
0, 1, -1, 1, -1, 1,
0, -1, 1, -1, 1, -1, 1,
0, 1, -1, 1, -1, 1, -1, 1,
0, -1, 1, -1, 1, -1, 1, -1, 1,
0, 1, -1, 1, -1, 1, -1, 1, -1, 1
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CROSSREFS
| The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Cf. A033184, A000108.
Cf. A106566 (unsigned version), A059365
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...
Sequence in context: A011434 A147746 A059365 * A106566 A205574 A049244
Adjacent sequences: A099036 A099037 A099038 * A099040 A099041 A099042
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KEYWORD
| easy,sign,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 23 2004
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