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 A191397 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) having k DHU's (here U=(1,1), H=(1,0), and D=(1,-1)). 2
 1, 1, 2, 3, 6, 9, 1, 18, 2, 28, 7, 56, 14, 89, 37, 179, 72, 1, 289, 170, 3, 585, 326, 13, 956, 726, 34, 1948, 1380, 104, 3214, 2970, 250, 1, 6591, 5616, 659, 4, 10959, 11829, 1502, 20, 22609, 22300, 3647, 64, 37833, 46306, 8019, 220, 78486, 87154, 18495, 620, 1, 132037, 179222, 39648, 1804, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row n has 1+floor(n/5) entries. Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n). T(n,0) = A191398(n). Sum_{k>=0} k*T(n,k) = A191389(n-1). LINKS FORMULA G.f.: G(t,z) = 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)). EXAMPLE T(6,1)=2 because we have HU(DHU)D and U(DHU)DH, where U=(1,1), D=(1,-1), H=(1,0) (the DHU's are shown between parentheses). Triangle starts:    1;    1;    2;    3;    6;    9,  1;   18,  2;   28,  7;   56, 14; MAPLE G := 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 21 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 21 do seq(coeff(P[n], t, k), k = 0 .. floor((1/5)*n)) end do; # yields sequence in triangular form CROSSREFS Cf. A001405, A191389, A191398. Sequence in context: A199790 A088329 A193079 * A087494 A328843 A021426 Adjacent sequences:  A191394 A191395 A191396 * A191398 A191399 A191400 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 04 2011 STATUS approved

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Last modified January 22 01:28 EST 2022. Contains 350481 sequences. (Running on oeis4.)