Search: id:A114503
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%I A114503
%S A114503 1,1,0,1,1,2,1,0,1,2,4,4,2,1,0,1,5,10,11,8,4,2,1,0,1,14,28,32,26,16,8,
               4,
%T A114503 2,1,0,1,42,84,98,84,57,32,16,8,4,2,1,0,1,132,264,312,276,198,120,64,32,
%U A114503 16,8,4,2,1,0,1,429,858,1023,924,687,438,247,128,64,32,16,8,4,2,1,0,1
%N A114503 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength 
               n for which height of first peak + height of last peak = k (n>=1; 
               2<=k<=2n).
%C A114503 Row n has 2n-1 terms. Column 2 yields the Catalan numbers (A000108). 
               T(n,3)=2T(n,2) (n>=3). Sum(kT(n,k),k=2..2n)=2[Catalan(n+1)-Catalan(n)] 
               (A071721). The trivariate g.f., with z marking semilength, t marking 
               height of the first peak and s marking height of the last peak, is 
               G = (1-tzC-szC+tsz^2*C^2+tsz^2*C)/[(1-tzC)(1-szC)(1-tsz)]-1.
%F A114503 G.f.=(1-2tzC+t^2*z^2*C^2+t^2*z^2*C)/[(1-tzC)^2*(1-t^2*z)]-1, where C=[1-sqrt(1-4z)]/
               (2z) is the Catalan function.
%e A114503 T(5,6)=4 because we have UUDUUUDDDD, UUUDUDUDDD, UUUDDUUDDD and UUUUDDDUDD, 
               where U=(1,1), D=(1,-1).
%e A114503 Triangle starts:
%e A114503 1;
%e A114503 1,0,1;
%e A114503 1,2,1,0,1;
%e A114503 2,4,4,2,1,0,1;
%e A114503 5,10,11,8,4,2,1,0,1;
%p A114503 C:=(1-sqrt(1-4*z))/2/z: g:=(1-2*t*z*C+t^2*z^2*C^2+t^2*z^2*C)/(1-t*z*C)^2/
               (1-t^2*z)-1: gser:=simplify(series(g,z=0,12)): for n from 1 to 10 
               do P[n]:=coeff(gser,z^n) od: for n from 1 to 10 do seq(coeff(P[n],
               t^j),j=2..2*n) od; # yields sequence in triangular form
%Y A114503 Cf. A000108, A071721.
%Y A114503 Sequence in context: A023444 A136868 A145895 this_sequence A103528 A138352 
               A129620
%Y A114503 Adjacent sequences: A114500 A114501 A114502 this_sequence A114504 A114505 
               A114506
%K A114503 nonn,tabf
%O A114503 1,6
%A A114503 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 02 2005

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