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A181327 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an even sum (0<=k<=floor(n/2)).

%I A181327 #14 May 27 2015 10:34:36
%S A181327 1,2,4,3,12,12,32,41,9,86,140,54,232,451,246,27,624,1416,1008,216,
%T A181327 1680,4357,3811,1215,81,4522,13192,13692,5832,810,12172,39455,47380,
%U A181327 25254,5400,243,32764,116820,159296,102024,29700,2916,88192,343029,523549
%N A181327 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an even sum (0<=k<=floor(n/2)).
%C A181327 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 A181327 Row n has 1+floor(n/2) entries.
%C A181327 The sum of entries in row n is A003480(n).
%C A181327 T(n,0) = A181329(n).
%C A181327 Sum(k*T(n,k), k>=0) = A181328(n).
%C A181327 For the statistic "number of column with an odd sum" see A181308.
%H A181327 Alois P. Heinz, <a href="/A181327/b181327.txt">Rows n = 0..200, flattened</a>
%H A181327 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 A181327 G.f.: G(t,z) = (1-z)^2*(1+z)^2/(1-2*z-2*z^2+z^4-3*t*z^2+t*z^4).
%F A181327 The g.f. of column k is z^{2k}*(1-z^2)^2*(3-z^2)^k/(1-2z-2z^2+z^4)^{k+1}
%F A181327 The g.f. H(t,s,z), where z marks size and t (s) marks number of columns with an odd (even) sum, is H=(1-z^2)^2/(1-2z^2+z^4-2tz-3sz^2+sz^4).
%e A181327 T(2,1) = 3 because we have (0 / 2), (1 / 1), and (2 / 0) (the 2-compositions are written as (top row / bottom row)).
%e A181327 Triangle starts:
%e A181327 1;
%e A181327 2;
%e A181327 4,    3;
%e A181327 12,  12;
%e A181327 32,  41,  9;
%e A181327 86, 140, 54;
%p A181327 G := (1-z^2)^2/(1-2*z-2*z^2+z^4-3*t*z^2+t*z^4): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%p A181327 # second Maple program:
%p A181327 b:= proc(n) option remember; `if`(n=0, 1,
%p A181327        expand(add(add(`if`(i=0 and j=0, 0, b(n-i-j)*
%p A181327        `if`(irem(i+j,2)=0, x, 1)), i=0..n-j), j=0..n)))
%p A181327     end:
%p A181327 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A181327 seq(T(n), n=0..15);  # _Alois P. Heinz_, Mar 16 2014
%t A181327 b[n_] := b[n] = If[n == 0, 1, Expand[Sum[Sum[If[i == 0 && j == 0, 0, b[n-i-j] * If[Mod[i+j, 2] == 0, x, 1]], {i, 0, n-j}], {j, 0, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, May 27 2015, after _Alois P. Heinz_ *)
%Y A181327 Cf. A003480, A181308, A181328, A181329.
%K A181327 nonn,tabf
%O A181327 0,2
%A A181327 _Emeric Deutsch_, Oct 13 2010