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A191320 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) with k HUs, where U=(1,1) and H=(1,0). 0
1, 1, 2, 2, 1, 4, 2, 4, 6, 9, 10, 1, 9, 23, 3, 23, 36, 11, 23, 77, 25, 1, 65, 118, 65, 4, 65, 249, 131, 17, 197, 380, 298, 48, 1, 197, 808, 566, 140, 5, 626, 1236, 1210, 336, 24, 626, 2665, 2230, 833, 80, 1, 2056, 4094, 4627, 1828, 259, 6, 2056, 8955, 8401, 4155, 711, 32, 6918, 13816, 17192, 8648, 1923, 122, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row n has 1 + floor(n/3) entries.
Sum of entries in row n is binomial(n,floor(n/2)) = A001405(n).
T(2*n,0) = T(2*n+1,0) = A014137(n) (partial sums of the Catalan numbers).
Sum_{k>=0}k*T(n,k) = A093387(n).
LINKS
FORMULA
G.f.: G(t,z) = 2/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)).
EXAMPLE
T(7,2)=3 because we have HHUDHUD, HUDHHUD, and HUDHUDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
1;
1;
2;
2, 1;
4, 2;
4, 6;
9, 10, 1;
9, 23, 3;
23, 36, 11;
MAPLE
G := 2/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
CROSSREFS
Sequence in context: A289500 A325276 A098804 * A180228 A212320 A197376
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 01 2011
STATUS
approved

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Last modified February 23 09:02 EST 2024. Contains 370275 sequences. (Running on oeis4.)