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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
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OFFSET
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0,8
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COMMENTS
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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 Deléham, May 31 2005
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LINKS
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L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
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FORMULA
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T(n, k) = (-1)^(n+k)*binomial(2*n-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 Deléham, May 31 2005
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EXAMPLE
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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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MATHEMATICA
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T[n_, k_]:= If[n == 0 && k == 0, 1, If[n == 0 && k > 0, 0, (-1)^(n + k)*Binomial[2*n - k - 1, n - k]*k/n]]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)
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PROG
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(PARI) {T(n, k) = if(n == 0 && k == 0, 1, if(n == 0 && k > 0, 0, (-1)^(n + k)*binomial(2*n - k - 1, n - k)*k/n))};
for(n=0, 15, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 31 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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