cp's OEIS Frontend

A062993 A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences.

%I A062993 #67 Jan 05 2025 19:51:36
%S A062993 1,1,1,2,1,1,5,3,1,1,14,12,4,1,1,42,55,22,5,1,1,132,273,140,35,6,1,1,
%T A062993 429,1428,969,285,51,7,1,1,1430,7752,7084,2530,506,70,8,1,1,4862,
%U A062993 43263,53820,23751,5481,819,92,9
%N A062993 A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences.
%C A062993 The column sequences (without leading zeros) appear in eq.(7.66), p. 347 of the Graham et al. reference, in Th. 0.3, p. 66, of Hilton and Pedersen reference, as first columns of the S-triangles in the Hoggatt and Bicknell reference and in eq. 5 of the Frey and Sellers reference. They are also called m-Raney (here m=k+2) or Fuss-Catalan sequences (see Graham et al. for reference). For the history and the name Pfaff-Fuss see Brown reference, p. 975. PF(n,m) := binomial(m*n+1,n)/(m*n+1), m >= 2.
%C A062993 Also called generalized Catalan numbers.
%D A062993 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.
%H A062993 O. Aichholzer, A. Asinowski, and T. Miltzow, <a href="http://arxiv.org/abs/1403.5546">Disjoint compatibility graph of non-crossing matchings of points in convex position</a>, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
%H A062993 Jean-Christophe Aval, <a href="http://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan numbers</a>, arXiv:0711.0906 [math.CO], 2007.
%H A062993 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>
%H A062993 W. G. Brown, <a href="http://www.jstor.org/stable/2313332">Historical note on a recurrent combinatorial problem</a>, Am. Math. Monthly 72 (1965) 973-977.
%H A062993 Sergio Caracciolo and Andrea Sportiello, <a href="https://doi.org/10.1007/s10955-009-9733-1">Spanning forests on random planar lattices</a>, J. Stat. Phys. 135, No. 5-6, 1063-1104 (2009).
%H A062993 CombOS - Combinatorial Object Server, <a href="http://combos.org/kary">Generate k-ary trees and dissections</a>
%H A062993 M. Dziemianczuk, <a href="http://dx.doi.org/10.1016/j.disc.2014.07.024">Enumerations of plane trees with multiple edges and Raney lattice paths</a>, Discrete Mathematics 337 (2014): 9-24.
%H A062993 D. D. Frey and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/papers.htm">Generalizing Bailey's generalization of the Catalan numbers</a>, The Fibonacci Quarterly, 39 (2001) 142-148.
%H A062993 P. Hilton and J. Pedersen, <a href="http://dx.doi.org/10.1007/BF03024089">Catalan Numbers, their generalization and their uses</a>, The Mathematical Intelligencer 13 (1991) 64-75.
%H A062993 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.
%H A062993 V. U. Pierce, <a href="http://arXiv.org/abs/math/0703160">Combinatoric results for graphical enumeration and the higher Catalan numbers</a>, arXiv:math/0703160 [math.CO], 2007.
%H A062993 J. H. Przytycki and A. S. Sikora, <a href="http://arxiv.org/abs/math/9811086">Polygon dissections and Euler, Fuss, Kirkman and Cayley numbers</a>, arXiv:math/9811086 [math.CO], 1998.
%H A062993 H. S. Snevily and D. B. West, <a href="http://arXiv.org/abs/math.CO/9802026">The Bricklayer Problem and the Strong Cycle Lemma</a>, arXiv:math/9802026 [math.CO], 1998.
%H A062993 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Young/young5.html">J. Int. Seq.</a>, Vol. 23 (2020), Article 20.9.1.
%F A062993 a(n, k) = binomial((k+2)*(n-k), n-k)/((k+1)*(n-k)+1) = PF(n-k, k+2) if n-k >= 0, otherwise 0.
%F A062993 G.f. for column k: A(k, x) := x^k*RootOf(_Z^(k+2)*x-_Z+1) (Maple notation, from ECS, see links for column sequences and Graham et al. reference eq.(5.59) p. 200 and p. 349).
%e A062993 The triangle a(n, k) begins:
%e A062993 n\k     0      1      2      3     4     5    6   7  8  9 10 ...
%e A062993 0:      1
%e A062993 1:      1      1
%e A062993 2:      2      1      1
%e A062993 3:      5      3      1      1
%e A062993 4:     14     12      4      1     1
%e A062993 5:     42     55     22      5     1     1
%e A062993 6:    132    273    140     35     6     1    1
%e A062993 7:    429   1428    969    285    51     7    1   1
%e A062993 8:   1430   7752   7084   2530   506    70    8   1  1
%e A062993 9:   4862  43263  53820  23751  5481   819   92   9  1  1
%e A062993 10: 16796 246675 420732 231880 62832 10472 1240 117 10  1  1
%e A062993 ... Reformatted by _Wolfdieter Lang_, Feb 06 2020
%t A062993 a[n_, k_] = Binomial[(k+2)*(n-k), n-k]/((k+1)*(n-k) + 1);
%t A062993 Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 53]]
%t A062993 (* _Jean-François Alcover_, May 27 2011, after formula *)
%Y A062993 Reflected version of A070914.
%Y A062993 Columns k=0..9 (without leading zeros) give sequences A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A059968, A230388.
%K A062993 nonn,easy,tabl
%O A062993 0,4
%A A062993 _Wolfdieter Lang_, Jul 12 2001