

A062993


A triangle (lower triangular matrix) composed of PfaffFuss (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
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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 Striangles in the Hoggatt and Bicknell reference and in eq. 5 of the Frey and Sellers reference. They are also called mRaney (here m=k+2) or FussCatalan sequences (see Graham et al. for reference). For the history and the name PfaffFuss 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. AddisonWesley, 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 noncrossing matchings of points in convex position, arXiv preprint arXiv:1403.5546, 2014
JeanChristophe Aval, Multivariate FussCatalan 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) 973977.
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 924.
D.D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142148.
P. Hilton and J. Pedersen, Catalan Numbers, their generalization and their uses, The Mathematical Intelligencer 13 (1991) 6475.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395405.
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)*(nk), nk)/((k+1)*(nk)+1) = PF(nk, k+2) if nk >= 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)*(nk), nk]/((k+1)*(nk) + 1);
Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 53]]
(* JeanFrançois 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



