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A191399 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of semilength n having k peak plateaux.

%I A191399 #15 Jul 17 2017 02:16:41
%S A191399 1,1,2,3,5,1,8,2,13,7,21,14,34,35,1,55,68,3,89,149,14,144,282,36,233,
%T A191399 576,114,1,377,1068,267,4,610,2088,711,23,987,3810,1566,72,1597,7229,
%U A191399 3771,272,1,2584,13024,7953,744,5,4181,24179,17922,2304,34,6765,43114,36594,5780,125
%N A191399 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of semilength n having k peak plateaux.
%C A191399 A dispersed Dyck paths of semilength n is a Motzkin path of length n with no (1,0)-steps at positive heights. A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.
%C A191399 Row n has 1+floor(n/4) entries.
%C A191399 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191399 T(n,0) = A000045(n+1) (Fibonacci numbers).
%C A191399 Sum_{k>=0} k*T(n,k) = A191319(n-2).
%F A191399 G.f.: G=G(t,z) satisfies (t*z^4-z^4-2*z^3+z^2+2*z-1)*G*(1+z*G)+1-z^2=0.
%e A191399 T(8,2)=1 because we have (UUDD)(UUDD), where U=(1,1) and D=(1,-1) (the peak plateaux are shown between parentheses).
%e A191399 Triangle starts:
%e A191399    1;
%e A191399    1;
%e A191399    2;
%e A191399    3;
%e A191399    5,  1;
%e A191399    8,  2;
%e A191399   13,  7;
%e A191399   21, 14;
%e A191399   34, 35,  1;
%p A191399 eq := (t*z^4-z^4-2*z^3+z^2+2*z-1)*G*(1+z*G)+1-z^2 = 0: g := RootOf(eq, G): Gser := simplify(series(g, z = 0, 23)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 19 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form
%Y A191399 Cf. A000045, A001405, A191319.
%K A191399 nonn,tabf
%O A191399 0,3
%A A191399 _Emeric Deutsch_, Jun 05 2011