A191397 -id:A191397 - cp's OEIS frontend

A191398 Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

1, 1, 2, 3, 6, 9, 18, 28, 56, 89, 179, 289, 585, 956, 1948, 3214, 6591, 10959, 22609, 37833, 78486, 132037, 275316, 465255, 974659, 1653418, 3478520, 5920569, 12504448, 21344348, 45240473, 77417309, 164624203, 282335973, 602163830, 1034757445, 2212959172, 3809387953, 8167344875

Offset: 0

Emeric Deutsch, Jun 04 2011

nonn

			a(5)=9 because among the 10 (=A001405(5)) dispersed Dyck paths of length 5 only UDHUD has a DHU.

G. C. Greubel, Table of n, a(n) for n = 0..1000

g := 2/(1-z-2*z^3+(1-z)*sqrt(1-4*z^2)); gser := series(g, z = 0, 41); seq(coeff(gser, z, n), n = 0 .. 38);

CoefficientList[Series[2/(1-x-2*x^3+(1-x)*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

x='x+O('x^50); Vec(2/(1-x-2*x^3+(1-x)*sqrt(1-4*x^2))) \\ G. C. Greubel, Mar 26 2017

a(n) = A191397(n,0).
G.f.: 2/(1-z-2*z^3+(1-z)*sqrt(1-4*z^2)).
a(n) ~ 2^(n+7/2) * (1+3*(-1)^n/49) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014
Conjecture: -(n+2)*(n-3)*a(n) +(3*n^2-3*n-14)*a(n-1) +2*(n^2-7*n+8)*a(n-2) +4*(-3*n^2+12*n-10)*a(n-3) +(7*n^2-31*n+38)*a(n-4) +4*a(n-5) +4*(n-2)^2*a(n-6)=0. - R. J. Mathar, Jun 14 2016

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).

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Jun 04 2011

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).

			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;

Cf. A001405, A191393, A191397.

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

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).

cp's OEIS Frontend

A191398 Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

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