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A101974 Triangle read by rows: number of Dyck paths of semilength n with k peaks before the first return (1<= k

%I A101974 #6 Jun 21 2019 05:25:42
%S A101974 1,2,4,1,9,4,1,23,11,7,1,65,27,28,11,1,197,66,87,62,16,1,626,170,239,
%T A101974 250,122,22,1,2056,471,627,829,630,219,29,1,6918,1398,1656,2448,2553,
%U A101974 1419,366,37,1,23714,4381,4554,6803,8813,6979,2917,578,46,1,82500,14282
%N A101974 Triangle read by rows: number of Dyck paths of semilength n with k peaks before the first return (1<= k <n).
%D A101974 E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
%F A101974 T(n, 1)=sum(c(i), i=0..n-1), T(n, k)=sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) for k>1, where c(i)=binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108);
%F A101974 G.f.=1+tzC(z)[1+r(t, z)], where C(z)=1+zC(z)^2 is the Catalan function and r(t, z)=z[1+r(t, z)][1+tr(t, z)] is the Narayana function.
%e A101974 T(4,2)=4 because we have U(UD)(UD)D|UD, U(UD)U(UD)DD|, UU(UD)D(UD)D| and
%e A101974 UU(UD)(UD)DD|, where U=(1,1), D=(1,-1) (the peaks before the first return | are shown between parentheses).
%e A101974      1
%e A101974        2
%e A101974      4      1
%e A101974      9      4      1
%e A101974     23     11      7      1
%e A101974     65     27     28     11      1
%e A101974    197     66     87     62     16      1
%e A101974    626    170    239    250    122     22      1
%e A101974   2056    471    627    829    630    219     29      1
%e A101974   6918   1398   1656   2448   2553   1419    366     37      1
%e A101974  23714   4381   4554   6803   8813   6979   2917    578     46      1
%e A101974  82500  14282  13231  18571  27362  28364  17206   5567    872     56      1
%p A101974 c:=n->binomial(2*n,n)/(n+1):
%p A101974 T:=proc(n,k) if k=1 then sum(c(i),i=0..n-1) else sum(c(j)*binomial(n-1-j,k-1)*binomial(n-1-j,k)/(n-1-j),j=0..n-2) fi end proc:
%p A101974 T(1,1);
%p A101974 for n from 1 to 12 do seq(T(n,k),k=1..n-1) od; # yields the sequence in triangular form
%Y A101974 Cf. A000108 (row sums), A014137 (column k=1), A014151 (column k=2), A101975.
%K A101974 nonn,tabf
%O A101974 1,2
%A A101974 _Emeric Deutsch_, Dec 22 2004