cp's OEIS Frontend

A181305 Number of increasing columns in all 2-compositions of n.

%I A181305 #14 May 11 2025 09:14:16
%S A181305 0,1,5,24,104,432,1736,6820,26332,100308,377996,1411844,5234428,
%T A181305 19285252,70670972,257766212,936336572,3388962884,12226547132,
%U A181305 43983439684,157814634684,564917186372,2017873643708,7193745818436
%N A181305 Number of increasing columns in all 2-compositions of n.
%C A181305 Also, number of odd entries in the top rows of all 2-compositions of 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 A181305 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 A181305 For the case of the even entries see A181337.
%H A181305 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.
%H A181305 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-12,-4,12,-4).
%F A181305 G.f.: z*(1-z)^2/((1+z)*(1-4*z+2*z^2)^2).
%F A181305 a(n) = Sum_{k=0..n} k*A181304(n,k).
%e A181305 a(1) = 1 because in the 2-compositions of 1, namely (0/1) and (1/0) we have only one increasing column (the 2-compositions are written as (top row / bottom row)).
%e A181305 a(2) = 5 because in (0/2), (1/1), (2,0), (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)) we have 0+1+0+1+1+2+0 = 5 odd entries.
%p A181305 g := z*(1-z)^2/((1+z)*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, k), k = 0 .. 27);
%Y A181305 Cf. A181304, A181337.
%K A181305 nonn,easy
%O A181305 0,3
%A A181305 _Emeric Deutsch_, Oct 13 2010
%E A181305 Edited by _N. J. A. Sloane_, Oct 15 2010