0,4

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..8 (without leading zeros) give sequences A000108 (Catalan), A001764, A002293, A002294-6, A007556, A062994, A062744.

Also called generalized Catalan numbers.

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

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

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, The Fibonacci Quarterly 14 (1976) 395-405.

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

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

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

H. S. Snevily and D. B. West, The Bricklayer Problem and the Strong Cycle Lemma

V. U. Pierce, Combinatoric results for graphical enumeration and the higher Catalan numbers

Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906 [math.CO]. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008]

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).

{1}; {1,1}; {2,1,1}; {5,3,1,1}; ...

Reflected version of A070914.

Sequence in context: A055818 A106240 A097615 this_sequence A105556 A078920 A117396

Adjacent sequences: A062990 A062991 A062992 this_sequence A062994 A062995 A062996

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 12 2001