0,2

a(n) is sum of the (flattened) list obtained by the iteration of: replace each integer k by the list 0,..k+1 on the starting value 0. Length of this list is catalan(n)or A000108. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Nov 11 2001

a(n-2) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 3 (cf. Zoran Sunik reference) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 07 2003

Number of standard tableaux of shape (n+2,n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 30 2004

a(n) = CatalanNumber[n+3] - 2*CatalanNumber[n+2]. Proof. From its definition as a convolution of Catalan numbers, a(n) counts lists of 4 Dyck paths of total size (semilength) = n. Connect the 4 paths by 3 upsteps (U) and append 3 downsteps (D). This is a reversible procedure. So a(n) is also the number of Dyck (n+3)-paths that end DDD (D for downstep). Let C(n) denote CatalanNumber[n] (A000108). Since C(n+3) is the total number of Dyck (n+3)-paths and C(n+2) is the number that end UD, we have (*) C(n+3) - C(n+2) is the number of Dyck (n+3)-paths that end DD. Also, (**) C(n+2) is the number of Dyck (n+3)-paths that end UDD (change the last D in a Dyck (n+2)-path to UDD). Subtracting (**) from (*) yields a(n) = C(n+3) - 2C(n+2) as claimed. - David Callan (callan(AT)stat.wisc.edu), Nov 21 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.

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.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. See Eq.(3).

L. W. Shapiro, A Catalan triangle. Discrete Math. 14 (1976), no. 1, 83-90.

W.-J. Woan, L. Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, 104 (1997), 926-931.

Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

T. D. Noe, Table of n, a(n) for n=0..100

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

a(n)= A033184(n+3, 4)= 4*binomial(2*n+1, n-1)/(n+3)= 2*n*A000108(n+1)/(n+3).

G.f.: c(x)^4 with c(x) g.f. of A000108 (Catalan).

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 14 2008: (Start)

Row sums of A145596. Column 4 of A033184. By specialising the identities for the row polynomials given in A145596 we obtain the results a(n) = sum {k = 0..n} (-1)^k*binomial(n+1,k+1)*a(k)*4^(n-k) and a(n) = sum {k = 0..floor(n/2)} binomial(n+1,2*k+1) * Catalan(k+1) * 2^(n-2*k). From the latter identity we can derive the congruences a(2n+1) = 0 (mod 4) and a(2n) = Catalan(n+1) (mod 4). It follows that a(n) is odd if and only if n = (2^m - 4) for some m >= 2.

(End)

Table[Plus@@Flatten[Nest[ #/.a_Integer:> Range[0, a+1]&, {0}, n]], {n, 0, 10}]

(PARI) a(n)=if(n<0, 0, n+=2; 2*binomial(2*n, n-2)/n) /* Michael Somos Jul 31 2005 */

T(n, n+4) for n=0, 1, 2, ..., array T as in A047072. Also a diagonal of A059365 and of A009766.

Cf. A001003.

A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Cf. A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392.

A145596 (row sums). [From Peter Bala (pbala(AT)toucansurf.com), Oct 14 2008]

Sequence in context: A092489 A094827 A094667 this_sequence A099376 A047048 A071745

Adjacent sequences: A002054 A002055 A002056 this_sequence A002058 A002059 A002060

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).