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A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries (0<=k<=n).

%I A181297 #12 May 11 2025 09:08:30
%S A181297 1,0,2,1,0,6,0,8,0,16,3,0,35,0,44,0,28,0,132,0,120,8,0,160,0,460,0,
%T A181297 328,0,92,0,748,0,1528,0,896,21,0,642,0,3117,0,4916,0,2448,0,290,0,
%U A181297 3552,0,12062,0,15456,0,6688,55,0,2380,0,17119,0,44318,0,47760,0,18272,0,888,0
%N A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries (0<=k<=n).
%C A181297 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 A181297 For the statistics "number of odd entries" see A181295.
%H A181297 G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="https://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 A181297 G.f.: G(t,z) = (1-z^2)^2/(1-3*z^2+z^4-2*s*z-2*s^2*z^2+s^2*z^4).
%F A181297 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.
%F A181297 Sum_{k=0..n} T(n,k) = A003480(n).
%F A181297 T(2*n-1,0) = 0.
%F A181297 T(2*n,0) = A000045(2*n) (Fibonacci numbers).
%F A181297 T(n,k) = 0 if n and k have opposite parities.
%F A181297 T(n,n) = A002605(n+1).
%F A181297 Sum_{k=0..n} k*T(n,k) = A181298(n).
%e A181297 T(2,2) = 6 because we have (0 / 2), (2 / 0), (1,0 / 0,1), (0,1 / 1,0), (1,1 / 0,0), (0,0 / 1,1) (the 2-compositions are written as (top row / bottom row)).
%e A181297 Triangle starts:
%e A181297   1;
%e A181297   0,2;
%e A181297   1,0,6;
%e A181297   0,8,0,16;
%e A181297   3,0,35,0,44;
%p A181297 G := (1-z^2)^2/(1-3*z^2+z^4-2*s*z-2*s^2*z^2+s^2*z^4): 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], s, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A181297 Cf. A003480, A000045, A181295, A181296, A181298.
%K A181297 nonn,tabl
%O A181297 0,3
%A A181297 _Emeric Deutsch_, Oct 12 2010