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A112307 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n-1).

%I A112307 #2 Mar 30 2012 17:36:06
%S A112307 1,1,1,1,2,2,1,4,6,3,1,9,16,12,4,1,23,44,39,20,5,1,65,128,123,76,30,6,
%T A112307 1,197,392,393,268,130,42,7,1,626,1250,1284,928,505,204,56,8,1,2056,
%U A112307 4110,4287,3216,1880,864,301,72,9,1,6918,13834,14583,11224,6885,3438,1379
%N A112307 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n-1).
%C A112307 Row sums are the Catalan numbers (A000108). T(n,0)=1 (paths have only one peak); The g.f. for column k is kz^(k+1)*c^k/(1-z), where c=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,1)=A014137(n-1); T(n,2)=2*A014138(n-3); T(n,3)=3*A001453(n-2); T(n,4)=4*A114277(n-5); Sum(k*T(n,k), k=0..n-1)=A112308(n-2).
%F A112307 G.f.=[(1-tzc)^2+tz^2*c]/[(1-z)(1-tzc)^2]-1, where c=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e A112307 T(4,1)=4 because we have UDUDUDUD, UDUDUUDD, UUDDUDUD and UUUDDDUD, where U=(1,1), D=(1,-1).
%e A112307 Triangle begins:
%e A112307 1;
%e A112307 1,1;
%e A112307 1,2,2;
%e A112307 1,4,6,3;
%e A112307 1,9,16,12,4;
%p A112307 G:=((1-t*z*c)^2+t*z^2*c)/(1-z)/(1-t*z*c)^2-1: c:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G,z=0,15)): for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 12 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form
%Y A112307 Cf. A000108, A014137, A014138, A001453, A114277, A112308.
%K A112307 nonn,tabl
%O A112307 1,5
%A A112307 _Emeric Deutsch_, Nov 30 2005