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Number of Dyck paths of semilength n having exactly two (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).
3

%I #7 Jun 11 2014 19:54:21

%S 1,1,2,12,69,98,180,1056,3967,18357,77685,264563,1245762,1915056,

%T 5303208,24548040,107835695,375494210,1898502240,4942470942,

%U 23489565822,104559681798,413327570240,1426320927138,6025235528016,19911812844324,87316285518504

%N Number of Dyck paths of semilength n having exactly two (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).

%H Alois P. Heinz, <a href="/A243771/b243771.txt">Table of n, a(n) for n = 2..350</a>

%e a(2) = 1: (UD)[UD].

%e a(3) = 1: (U[U)U]DDD.

%e a(4) = 2: U(UDD)U[UDD], UU(UDD)[UDD].

%e a(5) = 12: (UD[U)DU]UDDUD, (UD[U)DU]UUDDD, (UDU)UDD[UDU]D, (UDU)[UDU]DDUD, (UDU)[UDU]UDDD, (UDU)U[UDU]DDD, UUDD(UD[U)DU]D, U(UDU)DD[UDU]D, U(UD[U)DU]DDUD, U(UD[U)DU]UDDD, U(UDU)[UDU]DDD, UU(UD[U)DU]DDD.

%Y Column k=2 of A243752.

%Y Main diagonal of A243828.

%K nonn

%O 2,3

%A _Alois P. Heinz_, Jun 10 2014