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 A191395 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0)-steps at positive heights) for which the sum of the heights of its base pyramid is k. A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis. Here U=(1,1) and D=(1,-1). 1
 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 2, 5, 9, 4, 3, 6, 14, 12, 8, 9, 20, 25, 8, 13, 14, 27, 44, 28, 31, 29, 40, 70, 66, 16, 49, 54, 62, 104, 129, 64, 109, 115, 116, 159, 225, 168, 32, 170, 212, 217, 250, 363, 360, 144, 371, 430, 445, 444, 581, 681, 416, 64, 581, 772, 854, 820, 938, 1182, 968, 320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row n has 1+ceiling(n/2) entries. Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n). T(n,0) = A191393(n). Sum_{k>=0} k*T(n,k) = A191397(n). LINKS FORMULA G.f.: G=G(t,z) satisfies G = 1+z*G+z^2*G*(c+t/(1-t*z^2)-1/(1-z^2)), where c = (1-sqrt(1-4*z^2))/(2*z^2) (the Catalan function with argument z^2). EXAMPLE T(4,2)=2 because we have UDUD and UUDD, where U=(1,1), D=(1,-1), H=(1,0). Triangle starts:   1;   1;   1,  1;   1,  2;   1,  3,  2;   1,  4,  5;   2,  5,  9,  4;   3,  6, 14, 12;   8,  9, 20, 25,  8; MAPLE eq := G = 1+z*G+z^2*G*(c+t/(1-t*z^2)-1/(1-z^2)): c := ((1-sqrt(1-4*z^2))*1/2)/z^2: g := simplify(solve(eq, G)): gser := simplify(series(g, z = 0, 19)): for n from 0 to 15 do P[n] := sort(expand(coeff(gser, z, n))) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form CROSSREFS Cf. A001405, A191393, A191397. Sequence in context: A191318 A341315 A293600 * A183917 A181971 A104741 Adjacent sequences:  A191392 A191393 A191394 * A191396 A191397 A191398 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 04 2011 STATUS approved

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Last modified May 29 05:00 EDT 2022. Contains 354122 sequences. (Running on oeis4.)