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A191781 Triangle read by rows: T(n,k) is the number of length-n left factors of Dyck paths having length of the first ascent equal to k (1 <= k <= n).

%I A191781 #11 Nov 03 2019 19:42:07
%S A191781 1,1,1,1,1,1,2,2,1,1,3,3,2,1,1,6,6,4,2,1,1,10,10,7,4,2,1,1,20,20,14,8,
%T A191781 4,2,1,1,35,35,25,15,8,4,2,1,1,70,70,50,30,16,8,4,2,1,1,126,126,91,56,
%U A191781 31,16,8,4,2,1,1,252,252,182,112,62,32,16,8,4,2,1,1,462,462,336,210,119,63,32,16,8,4,2,1,1
%N A191781 Triangle read by rows: T(n,k) is the number of length-n left factors of Dyck paths having length of the first ascent equal to k (1 <= k <= n).
%C A191781 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191781 T(n,1) = A001405(n-2) (n>=1).
%C A191781 T(n,2) = A001405(n-2) (n>=2).
%C A191781 Sum_{k=1..n} k*T(n,k) = A191782(n).
%F A191781 G.f.: G(t,z) = (1 - z*c + t*z^3*c^2)/((1 - z*c)*(1 - t*z)*(1 - t*z^2)), where c = (1-sqrt(1 - 4*z^2))/(2*z^2).
%e A191781 T(5,2)=3 because we have UUDDU, UUDUD, and UUDUU, where U=(1,1) and D=(1,-1).
%e A191781 Triangle starts:
%e A191781   1;
%e A191781   1, 1;
%e A191781   1, 1, 1;
%e A191781   2, 2, 1, 1;
%e A191781   3, 3, 2, 1, 1;
%e A191781   6, 6, 4, 2, 1, 1;
%p A191781 c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z*c+t*z^3*c^2)/((1-z*c)*(1-t*z)*(1-t*z^2*c)): Gser := simplify(series(G, z = 0, 17)): for n to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 13 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form
%Y A191781 Cf. A001405, A191782.
%K A191781 nonn,tabl
%O A191781 1,7
%A A191781 _Emeric Deutsch_, Jun 18 2011