cp's OEIS Frontend

A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even 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 A181336 #2 Mar 30 2012 17:36:24
%S A181336 1,1,1,2,4,1,5,11,7,1,11,31,29,10,1,25,83,102,56,13,1,56,217,329,245,
%T A181336 92,16,1,126,556,1000,938,487,137,19,1,283,1403,2917,3292,2180,855,
%U A181336 191,22,1,636,3498,8247,10865,8740,4406,1376,254,25,1,1429,8636,22756,34248
%N A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even 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 A181336 The sum of entries in row n is A003480(n).
%C A181336 T(n,0)=A006054(n+1) (n>=1).
%C A181336 Sum(k*T(n,k), k>=0)=A181337(n).
%C A181336 For the statistic "number of odd entries in the top row" see A181304.
%D A181336 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 A181336 G.f.=G(s,z)=(1+z)(1-z)^2/[1-2z-z^2+z^3-sz(1+z-z^2)].
%F A181336 The g.f. of column k is z^k*(1+z)(1-z)^2*(1+z-z^2)^k/(1-2z-z^2+z^3)^{k+1} (we have a Riordan array).
%F A181336 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 A181336 T(2,1)=4 because we have (0/2), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
%e A181336 Triangle starts:
%e A181336 1;
%e A181336 1,1;
%e A181336 2,4,1;
%e A181336 5,11,7,1;
%e A181336 11,31,29,10,1;
%e A181336 25,83,102,56,13,1;
%p A181336 G := (1+z)*(1-z)^2/(1-2*z-z^2+z^3-s*z*(1+z-z^2)): Gser := simplify(series(G, z = 0, 13)): 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], s, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A181336 Cf. A003480, A006054, A181304, A181337,
%K A181336 nonn,tabl
%O A181336 0,4
%A A181336 _Emeric Deutsch_, Oct 14 2010