A191795 - OEIS

%I #8 Nov 25 2019 01:00:26

%S 1,1,2,3,5,1,7,3,11,9,15,19,1,23,42,5,31,77,18,47,150,54,1,63,255,137,

%T 7,95,464,333,32,127,753,720,115,1,191,1314,1558,360,9,255,2067,3067,

%U 996,50,383,3508,6167,2597,214,1,511,5397,11410,6207,774,11,767,8982,21820,14485,2494,72

%N Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k DUU's, where U=(1,1) and D=(1,-1).

%C Row n >= 1 contains ceiling(n/3) entries.

%C Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

%C T(n,0) = A052955(n-1).

%C Sum_{k>=0} k*T(n,k) = A191796(n).

%F G.f.: G(t,z) = 1 - (1-C-z*C)/(1-z+t*z-t*z*C), where C=C(t,z) is given by t*z^2*C^2 - (1-2*z^2+2*t*z^2)*C + 1-z^2+t*z^2 = 0.

%e T(5,1)=3 because we have U(DUU)D, U(DUU)U, and UU(DUU), where U=(1,1) and D=(1,-1) (the DUU's are shown between parentheses).

%e Triangle starts:

%e    1;

%e    1;

%e    2;

%e    3;

%e    5,  1;

%e    7,  3;

%e   11,  9;

%e   15, 19, 1;

%e   23, 42, 5;

%p eq := t*z^2*C^2-(1-2*z^2+2*t*z^2)*C+1-z^2+t*z^2 = 0: C := RootOf(eq, C): G := 1-(1-C-z*C)/(1-z+t*z-t*z*C): Gser := simplify(series(G, z = 0, 23)): for n from 0 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 18 do seq(coeff(P[n], t, k), k = 0 .. ceil((1/3)*n)-1) end do; # yields sequence in triangular form

%Y Cf. A001405, A052955, A191796.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Jun 18 2011