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A135331 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k DUUU's starting at level 1.

%I A135331 #8 Oct 27 2015 04:44:08
%S A135331 1,1,2,5,13,1,36,6,105,27,320,108,1,1011,409,10,3289,1508,65,10957,
%T A135331 5491,347,1,37216,19898,1658,14,128435,72063,7395,119,449142,261436,
%U A135331 31527,794,1,1588228,951258,130353,4583,18
%N A135331 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k DUUU's starting at level 1.
%C A135331 The formula for T(n,k)=l_{n,k} in the reference (p. 2919) does not appear to work (a typo is possible). - _Emeric Deutsch_, Dec 14 2007
%C A135331 Row 0 has 1 term; row n (n>=1) has floor((n+2)/3) terms. Row sums are the Catalan numbers (A000108). Column 0 yields A135337. - _Emeric Deutsch_, Dec 14 2007
%H A135331 A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.
%F A135331 G.f.: G(t,z)=1+zC^2/[1+(1-t)z^3*C^4], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - _Emeric Deutsch_, Dec 14 2007
%e A135331 Triangle begins:
%e A135331 1
%e A135331 1
%e A135331 2
%e A135331 5
%e A135331 13 1
%e A135331 36 6
%e A135331 105 27
%e A135331 320 108 1
%e A135331 1011 409 10
%e A135331 3289 1508 65
%e A135331 10957 5491 347 1
%e A135331 ...
%e A135331 T(5,1)=6 because we have U(DUUU)UDDDD, U(DUUU)DUDDD, U(DUUU)DDUDD, U(DUUU)DDDUD, UDU(DUUU)DDD and UUD(DUUU)DDD (the DUUU's starting at level 1 are shown between parentheses).
%p A135331 G:=1+z*C^2/(1+(1-t)*z^3*C^4): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=simplify(series(G,z=0,16)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) end do: 1; for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((n-1)*1/3)) end do; # yields sequence in triangular form; _Emeric Deutsch_, Dec 14 2007
%Y A135331 Cf. A000108, A135337.
%K A135331 nonn,tabf
%O A135331 0,3
%A A135331 _N. J. A. Sloane_, Dec 07 2007
%E A135331 More terms from _Emeric Deutsch_, Dec 14 2007