|
%I #32 May 21 2022 08:41:34
%S 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,
%T 1,0,132,-132,90,-48,20,-6,1,0,-429,429,-297,165,-75,27,-7,1,0,1430,
%U -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
%N Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.
%C 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.
%C 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
%H G. C. Greubel, <a href="/A099039/b099039.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H George Beck and Karl Dilcher, <a href="https://arxiv.org/abs/2106.10400">A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence</a>, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.
%H F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.
%H D. Callan, <a href="https://arxiv.org/abs/math/0211380">A recursive bijective approach to counting permutations containing 3-letter patterns</a>, arXiv:math/0211380 [math.CO], 2002.
%H E. Deutsch, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00371-9">Dyck path enumeration</a>, Discrete Math., 204, 1999, 167-202.
%H R. K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%H A. Robertson, D. Saracino and D. Zeilberger, <a href="https://arxiv.org/abs/math/0203033">Refined restricted permutations</a>, arXiv:math/0203033 [math.CO], 2002.
%H L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan Group</a>, Discrete Appl. Maths. 34 (1991) 229-239.
%F 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
%e Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1}, ...
%e Triangle begins
%e 1;
%e 0, 1;
%e 0, -1, 1;
%e 0, 2, -2, 1;
%e 0, -5, 5, -3, 1;
%e 0, 14, -14, 9, -4, 1;
%e 0, -42, 42, -28, 14, -5, 1;
%e 0, 132, -132, 90, -48, 20, -6, 1;
%e 0, -429, 429, -297, 165, -75, 27, -7, 1;
%e Production matrix is
%e 0, 1,
%e 0, -1, 1,
%e 0, 1, -1, 1,
%e 0, -1, 1, -1, 1,
%e 0, 1, -1, 1, -1, 1,
%e 0, -1, 1, -1, 1, -1, 1,
%e 0, 1, -1, 1, -1, 1, -1, 1,
%e 0, -1, 1, -1, 1, -1, 1, -1, 1,
%e 0, 1, -1, 1, -1, 1, -1, 1, -1, 1
%t 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 *)
%o (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))};
%o for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 31 2017
%Y The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
%Y Cf. A033184, A000108.
%Y Cf. A106566 (unsigned version), A059365
%Y The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...
%K easy,sign,tabl
%O 0,8
%A _Paul Barry_, Sep 23 2004
|
