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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

%I

%S 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,

%T 31,29,40,70,66,16,49,54,62,104,129,64,109,115,116,159,225,168,32,170,

%U 212,217,250,363,360,144,371,430,445,444,581,681,416,64,581,772,854,820,938,1182,968,320

%N 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).

%C Row n has 1+ceiling(n/2) entries.

%C Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

%C T(n,0) = A191393(n).

%C Sum_{k>=0} k*T(n,k) = A191397(n).

%F 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).

%e T(4,2)=2 because we have UDUD and UUDD, where U=(1,1), D=(1,-1), H=(1,0).

%e Triangle starts:

%e 1;

%e 1;

%e 1, 1;

%e 1, 2;

%e 1, 3, 2;

%e 1, 4, 5;

%e 2, 5, 9, 4;

%e 3, 6, 14, 12;

%e 8, 9, 20, 25, 8;

%p 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

%Y Cf. A001405, A191393, A191397.

%K nonn,tabf

%O 0,6

%A _Emeric Deutsch_, Jun 04 2011

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Last modified January 23 07:07 EST 2020. Contains 331168 sequences. (Running on oeis4.)