cp's OEIS Frontend

A181299 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)).

%I A181299 #22 May 11 2025 09:14:04
%S A181299 1,2,6,1,20,4,64,17,1,206,68,6,662,261,32,1,2128,976,152,8,6840,3577,
%T A181299 675,51,1,21986,12912,2860,280,10,70670,46049,11704,1406,74,1,227156,
%U A181299 162628,46632,6632,460,12,730152,569705,181877,29866,2570,101,1,2346942
%N A181299 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)).
%C A181299 A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
%C A181299 Row n contains 1+floor(n/2) entries.
%H A181299 G. Castiglione, A. Frosini, E. Munarini, A. Restivo, and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European J. Combin. 28 (2007), no. 6, 1724-1741.
%F A181299 G.f.: G(t,z) = (1+z)*(1-z)^2/(1-3*z-z^2+z^3-t*(1-z)*z^2).
%F A181299 Sum_{k>=0} T(n,k) = A003480(n).
%F A181299 T(n,0) = A181301(n).
%F A181299 Sum_{k>=0} k*T(n,k) = A181300(n).
%e A181299 T(3,1) = 4 because we have (1,1/1,0), (1,0/1,1), (1,1/0,1), (0,1/1,1) (the 2-compositions are written as (top row/bottom row)).
%e A181299 Triangle starts:
%e A181299   1;
%e A181299   2;
%e A181299   6,1;
%e A181299   20,4;
%e A181299   64,17,1;
%p A181299 G := (1-z)^2*(1+z)/(1-3*z-z^2+z^3-t*z^2*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A181299 Cf. A003480, A181300, A181301.
%K A181299 nonn,tabf
%O A181299 0,2
%A A181299 _Emeric Deutsch_, Oct 12 2010