0,3

Number of peaks at level 1 in all Schroeder paths of semilength n (n>=1). Example: a(2)=4 because in the six Schroeder paths of semilength two, HH, H(UD), (UD)H, (UD)(UD), UHD and UUDD (where H=(2,0), U=(1,1), D=(1,-1)), we have four peaks at level 1 (shown between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003

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

G. Kreweras, Sur les hi\'{e}rarchies de segments, Cahiers Bureau Universitaire Recherche Op\'{e}rationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.

Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.

All listed terms satisfy the recurrence a(1)=1 and, for n>1, a(n)=4a(n-1)+Sum[a(k)a(n-k-1), k= 2, ..., n-2] - John W. Layman (layman(AT)math.vt.edu), Feb 23 2001

a(0)=1, for n>0: a(n)=Sum(Sum a(i)a(j-i), (i=0, .., j))(n-j), (j=0, .., n). G.f.: A(x)= (1/(2x))((1 - x)^2 - Sqrt[(1 - x)^4 - 4x(1 - x)^2]) - Mario Catalani (mario.catalani(AT)unito.it), Jun 19 2003

a(n)=0^n+sum{k=0..n-1,C(n+k,2k+1)*A000108(k)}; [From Paul Barry (pbarry(AT)wit.ie), Feb 01 2009]

d[n_] := d[n] = Sum[Sum[d[i]d[j - i], {i, 0, j}](n - j), {j, 0, n}]; d[0] = 1; Table[d[n], {n, 0, 26}]

First differences of A006318. Second diagonal of A033877.

Sequence in context: A089979 A128730 A151243 this_sequence A059606 A000303 A144316

Adjacent sequences: A006316 A006317 A006318 this_sequence A006320 A006321 A006322

nonn,easy

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