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A006319
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Royal paths in a lattice (convolution of A006318).
(Formerly M3521)
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8
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1, 1, 4, 16, 68, 304, 1412, 6752, 33028, 164512, 831620, 4255728, 22004292, 114781008, 603308292, 3192216000, 16989553668, 90890869312, 488500827908, 2636405463248, 14281895003716, 77631035881072, 423282220216964
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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REFERENCES
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| 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]
G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x)^2 (continued fraction); more generally g.f. C(x/(1-x)^2) where C(x) is the g.f. for the Catalan numbers (A000108). [Joerg Arndt, Mar 18 2011]
a(n) is the sum of top row terms in M^(n-1), M = an infinite square production matrix as follows:
2, 2, 0, 0, 0, 0,...
1, 1, 2, 0, 0, 0,...
1, 1, 1, 2, 0, 0,...
1, 1, 1, 1, 2, 0,...
1, 1, 1, 1, 1, 2,...
... - Gary W. Adamson, Aug 23 2011
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EXAMPLE
| a(4) = 68 since the top row of M^3 = (22, 22, 16, 8, 0, 0, 0,...); where 68 = (22 + 22 + 16 + 8)
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MATHEMATICA
| 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}]
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CROSSREFS
| First differences of A006318. Second diagonal of A033877.
Sequence in context: A179191 A128730 A151243 * A202020 A059606 A000303
Adjacent sequences: A006316 A006317 A006318 * A006320 A006321 A006322
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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