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A181332 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k nonzero 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.

%I A181332 #11 Nov 26 2013 11:24:10
%S A181332 1,1,1,2,4,1,4,12,7,1,8,32,31,10,1,16,80,111,59,13,1,32,192,351,268,
%T A181332 96,16,1,64,448,1023,1037,530,142,19,1,128,1024,2815,3598,2435,924,
%U A181332 197,22,1,256,2304,7423,11535,9843,4923,1477,261,25,1,512,5120,18943,34832
%N A181332 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k nonzero 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 A181332 The sum of entries in row n is A003480(n).
%C A181332 T(n,1) = A001787(n).
%C A181332 T(n,2) = A055580(n-2) (n>=2).
%C A181332 T(n,3) = A055586(n-3) (n>=3).
%C A181332 Sum(k*T(n,k), k>=0) = A054146(n).
%H A181332 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 A181332 T(n,k) = sum(2^j*binomial(k+j,k)*binomial(n-2-j,k-2), j=0..n-k).
%F A181332 G.f.: G(t,x) = (1-x)^2/(1-3*x+2*x^2-t*x).
%F A181332 The g.f. of column k is x^k/((1-2*x)^(k+1)*(1-x)^(k-1)) (we have a Riordan array).
%F A181332 T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), with T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=2, T(2,1)=4, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham, Nov 26 2013
%e A181332 T(2,1)=4 because we have (1/1), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
%e A181332 Triangle starts:
%e A181332 1;
%e A181332 1,1;
%e A181332 2,4,1;
%e A181332 4,12,7,1;
%e A181332 8,32,31,10,1;
%e A181332 16,80,111,59,13,1;
%p A181332 T := proc (n, k) options operator, arrow: sum(2^j*binomial(k+j, k)*binomial(n-j-2, k-2), j = 0 .. n-k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A181332 Cf. A003480, A001787, A055580, A055586, A181330, A181332.
%K A181332 nonn,tabl
%O A181332 0,4
%A A181332 _Emeric Deutsch_, Oct 13 2010