login
Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with increasing entries (0<=k<=n). 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.
3

%I #3 May 01 2013 21:13:45

%S 1,1,1,3,3,1,7,11,5,1,18,33,23,7,1,44,100,87,39,9,1,110,288,310,177,

%T 59,11,1,272,820,1036,728,311,83,13,1,676,2288,3338,2768,1450,497,111,

%U 15,1,1676,6316,10416,9976,6172,2588,743,143,17,1,4160,17244,31752,34448

%N Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with increasing entries (0<=k<=n). 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 Also, triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd entries in the top row. 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 The sum of entries in row n is A003480(n).

%C For the statistic "number of even entries in the top row" see A181336.

%C T(n,0)=A181306(n).

%C Sum(k*T(n,k),k>=0)=A181305(n).

%D G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.

%F G.f. = G(t,z)=(1+z)(1-z)^2/[1-(2+t)z-2z^2+2z^3].

%F G.f. for column k is z^k*(1+z)(1-z)^2/(1-2z-2z^2+2z^3)^{k+1} (we have a Riordan array).

%F The g.f. H=H(t,s,z), where z marks size and t (s) marks odd (even) entries in the top row, is given by H = (1+z)(1-z)^2/[(1+z)(1-z)^2-(t+s)z-sz^2*(1-z)].

%e T(2,1)=3 because we have (0/2), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row).

%e Alternatively, T(2,1)=3 because we have (1/1), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).

%e Triangle starts:

%e 1;

%e 1,1;

%e 3,3,1;

%e 7,11,5,1;

%e 18,33,23,7,1;

%e 44,100,87,39,9,1;

%p G := (1+z)*(1-z)^2/(1-(2+t)*z-2*z^2+2*z^3): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

%Y Cf. A003480, A181305, A181306, A181336

%K nonn,tabl

%O 0,4

%A _Emeric Deutsch_, Oct 13 2010

%E Edited by _N. J. A. Sloane_, Oct 15 2010