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A181295 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd 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.

%I A181295 #2 Mar 30 2012 17:36:24
%S A181295 1,0,2,2,0,5,0,12,0,12,7,0,46,0,29,0,58,0,152,0,70,24,0,297,0,466,0,
%T A181295 169,0,256,0,1236,0,1364,0,408,82,0,1632,0,4575,0,3870,0,985,0,1072,0,
%U A181295 8160,0,15702,0,10736,0,2378,280,0,8160,0,35320,0,51121,0,29282,0,5741,0
%N A181295 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd 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 A181295 The sum of entries in row n is A003480(n).
%C A181295 T(2n-1,0)=0.
%C A181295 T(2n,0)=A003480(n).
%C A181295 T(n,k)=0 if n and k have opposite parities.
%C A181295 T(n,n)=A000129(n+1) (the Pell numbers).
%C A181295 Sum(k*T(n,k),k=0..n)=A181296.
%C A181295 For the statistics "number of even entries" see A181297.
%D A181295 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 A181295 G.f.=G(t,z)=(1-z^2)^2/(1-4z^2+2z^4-2tz-t^2*z^2).
%F A181295 The g.f. H(t,s,z), where z marks the size of the 2-composition and t (s) marks the number of odd (even) entries, is H=1/(1-h), where h=z(t+sz)(2s+tz-sz^2)/(1-z^2)^2.
%e A181295 T(2,2)=5 because we have (1/1),(1,0/0,1),(0,1/1,0),(1,1/0,0), and (0,0/1,1); the 2-compositions are written as (top row / bottom row).
%e A181295 Triangle starts:
%e A181295 1;
%e A181295 0,2;
%e A181295 2,0,5;
%e A181295 0,12,0,12;
%e A181295 7,0,46,0,29;
%p A181295 G := (1-z^2)^2/(1-4*z^2+2*z^4-2*t*z-t^2*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A181295 Cf. A003480, A000129, A181296, A181297, A181298,
%K A181295 nonn,tabl
%O A181295 0,3
%A A181295 _Emeric Deutsch_, Oct 12 2010