This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181304 #13 May 11 2025 09:14:08
%S A181304 1,1,1,3,3,1,7,11,5,1,18,33,23,7,1,44,100,87,39,9,1,110,288,310,177,
%T A181304 59,11,1,272,820,1036,728,311,83,13,1,676,2288,3338,2768,1450,497,111,
%U A181304 15,1,1676,6316,10416,9976,6172,2588,743,143,17,1,4160,17244,31752,34448
%N A181304 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with increasing entries (0<=k<=n).
%C A181304 Also, triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd 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 A181304 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 A181304 The sum of entries in row n is A003480(n).
%C A181304 For the statistic "number of even entries in the top row" see A181336.
%H A181304 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 A181304 T(n,0) = A181306(n).
%F A181304 Sum_{k>=0} k*T(n,k) = A181305(n).
%F A181304 G.f.: G(t,z) = (1+z)*(1-z)^2/(1-(2+t)*z-2*z^2+2*z^3).
%F A181304 G.f. for column k: z^k*(1+z)*(1-z)^2/(1-2*z-2*z^2+2*z^3)^(k+1) (we have a Riordan array).
%F A181304 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 A181304 T(2,1) = 3 because we have (0/2), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
%e A181304 Alternatively, T(2,1) = 3 because we have (1/1), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
%e A181304 Triangle starts:
%e A181304 1;
%e A181304 1,1;
%e A181304 3,3,1;
%e A181304 7,11,5,1;
%e A181304 18,33,23,7,1;
%e A181304 44,100,87,39,9,1;
%p A181304 G := (1+z)*(1-z)^2/(1-(2+t)*z-2*z^2+2*z^3): Gser := simplify(series(G, z = 0, 15)): 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], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A181304 Cf. A003480, A181305, A181306, A181336.
%K A181304 nonn,tabl
%O A181304 0,4
%A A181304 _Emeric Deutsch_, Oct 13 2010
%E A181304 Edited by _N. J. A. Sloane_, Oct 15 2010