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A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers. 17
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; text; internal format)
OFFSET
0,8
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 Deléham, May 31 2005
LINKS
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
FORMULA
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
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Cf. A106566 (unsigned version), A059365
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Sequence in context: A147746 A059365 A106566 * A205574 A049244 A110281
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Sep 23 2004
STATUS
approved

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)