0,3

Number of peaks at even level in all Dyck paths of semilength n+1. Example: a(2)=4 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even level are shown by *. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003

Also number of long ascents (i.e. ascents of length at least two) in all Dyck paths of semilength n+1. Example: a(2)=4 because in the five Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and (UUU)DDD, we have four long ascents (shown between parentheses). Here U=(1,1) and D=(1,-1). Also number of branch nodes (i.e. vertices of outdegree at least two) in all ordered trees with n+1 edges. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=1. Example: For n=2 these are the paths EENN, ENEN, ENNE and NEEN. - Herbert Kociemba (kociemba(AT)t-online.de), May 23 2004

Narayana transform (A001263) of [1, 3, 5, 7, 9,...] = (1, 4, 15, 56, 210,...). Row sums of triangles A136534 and A136536. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 04 2008

Starting with offset 1 = the Catalan sequence starting (1, 2, 5, 14,...) convolved with A000984: (1, 2, 6, 20,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]

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).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.

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..200

Milan Janjic, Two Enumerative Functions

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

G.f.: x*diff(c(x), x), c(x) = Catalan g.f. (wolfdieter.lang(AT)physik.uni-karlsruhe.de).

Convolution of A001700( central binomial of odd order) and A000108 (Catalan): a(n+1)=sum(C(k)*binomial(2*(n-k)+1, n-k), k=0..n), C(k): Catalan [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) ].

E.g.f.: exp(2x) I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 08 2002

a(n)=sum{k=0..n, C(n, k)C(n, k+1) } - Paul Barry (pbarry(AT)wit.ie), May 15 2003

a(n)=sum(binomial(i+n-1, n), i=1..n).

G.f.=[1-2x-sqrt(1-4x)]/[2xsqrt(1-4x)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003

A092956/(n!) - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 16 2004

a(n)=C(2n, n)-C(n); - Paul Barry (pbarry(AT)wit.ie), Apr 21 2005

a(n)=(1/(2*pi))*Int(x^n*(x-2)/sqrt(x(4-x)),x,0,4) is the moment sequence representation; - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007

Row sums of triangle A132812 starting (1, 4, 15, 56, 210,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2007

Starting (1, 4, 15, 56, 210,...) gives the binomial transform of A025566 starting (1, 3, 8, 22, 61, 171,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2007

[seq(binomial(2*n, n)/(n+1)*n, n=0..30)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2006

a:=n->add(binomial(2*n, n)/(n+1), k=1..n): seq(a(n), n=0..24); - ZerinvaryLajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007

(PARI) a(n)=if(n<1, 0, (2*n)!/(n+1)!/(n-1)!)

(Mupad) combinat::catalan(n) *binomial(n, 1) $ n = 0..24 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2007

n*C(n), C(n)=Catalan A000108. Cf. A000984.

Diagonal 3 of triangle A100257.

First differences are in A076540.

Cf. A000108, A000984, A002378.

Cf. A025566, A132812.

Adjacent sequences: A001788 A001789 A001790 this_sequence A001792 A001793 A001794

Sequence in context: A026030 A047038 A158500 this_sequence A047128 A087438 A131497

nonn,easy,nice

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 08 2000

More terms from ZerinvaryLajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007