A191305 - OEIS

%I #10 Jul 17 2017 03:53:56

%S 1,1,1,1,1,2,2,3,1,3,4,3,6,7,6,1,9,12,10,4,18,23,18,10,1,28,40,33,20,

%T 5,57,76,64,39,15,1,91,134,120,76,35,6,187,257,231,152,75,21,1,304,

%U 460,433,300,156,56,7,629,888,834,595,325,132,28,1,1037,1606,1572,1164,670,294,84,8,2157,3115,3035,2292,1375,642,217,36,1

%N Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k hills (i.e., peaks at height 1).

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

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

%C Sum_{k>=0}k*T(n,k) = A045621(n-2).

%F G.f.: G=G(t,z) satisfies G = 1+z*G + z^2*G(C-1+t), where C=1+z^2*C^2 (and G=2/(1-2*z+2*z^2-2*t*z^2+sqrt(1-4*z^2)), see Maple program).

%e T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).

%e Triangle starts:

%e   1;

%e   1;

%e   1,  1;

%e   1,  2;

%e   2,  3,  1;

%e   3,  4,  3;

%e   6,  7,  6,  1;

%e   9, 12, 10,  4;

%p G := 2/(1-2*z+2*z^2-2*t*z^2+sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

%Y Cf. A001405, A045621.

%K nonn,tabf

%O 0,6

%A _Emeric Deutsch_, May 30 2011