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A062993 A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences. 16
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, 429, 1428, 969, 285, 51, 7, 1, 1, 1430, 7752, 7084, 2530, 506, 70, 8, 1, 1, 4862, 43263, 53820, 23751, 5481, 819, 92, 9 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

Columns k=0..9 (without leading zeros) give sequences A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A062744, A230388.

Also called generalized Catalan numbers.

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.

LINKS

Table of n, a(n) for n=0..52.

O. Aichholzer, A. Asinowski, T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546, 2014

Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906 [math.CO].

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps

W. G. Brown, Historical note on a recurrent combinatorial problem, Am. Math. Monthly 72 (1965) 973-977.

M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.

D.D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.

P. Hilton and J. Pedersen, Catalan Numbers, their generalization and their uses, The Mathematical Intelligencer 13 (1991) 64-75.

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

V. U. Pierce, Combinatoric results for graphical enumeration and the higher Catalan numbers, arXiv:math/0703160 [math.CO], 2007.

J. H. Przytycki and A. S. Sikora, Polygon dissections and Euler, Fuss, Kirkman and Cayley numbers, arXiv:math/9811086 [math.CO], 1998.

H. S. Snevily and D. B. West, The Bricklayer Problem and the Strong Cycle Lemma, arXiv:math/9802026 [math.CO], 1998.

FORMULA

a(n, k)= binomial((k+2)*(n-k), n-k)/((k+1)*(n-k)+1) = PF(n-k, k+2) if n-k >= 0, else 0.

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).

EXAMPLE

1;

1,1;

2,1,1;

5,3,1,1;

...

MATHEMATICA

a[n_, k_] = Binomial[(k+2)*(n-k), n-k]/((k+1)*(n-k) + 1);

Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 53]]

(* Jean-Fran├žois Alcover, May 27 2011, after formula *)

CROSSREFS

Reflected version of A070914.

Sequence in context: A106240 A097615 A288386 * A105556 A078920 A186020

Adjacent sequences:  A062990 A062991 A062992 * A062994 A062995 A062996

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jul 12 2001

STATUS

approved

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Last modified October 22 21:09 EDT 2018. Contains 316505 sequences. (Running on oeis4.)