

A181304


Triangle read by rows: T(n,k) is the number of 2compositions of n having k columns with increasing entries (0<=k<=n). A 2composition 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.


3



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, 59, 11, 1, 272, 820, 1036, 728, 311, 83, 13, 1, 676, 2288, 3338, 2768, 1450, 497, 111, 15, 1, 1676, 6316, 10416, 9976, 6172, 2588, 743, 143, 17, 1, 4160, 17244, 31752, 34448
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

Also, triangle read by rows: T(n,k) is the number of 2compositions of n having k odd entries in the top row. A 2composition 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.
The sum of entries in row n is A003480(n).
For the statistic "number of even entries in the top row" see A181336.
T(n,0)=A181306(n).
Sum(k*T(n,k),k>=0)=A181305(n).


REFERENCES

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of Lconvex polyominoes, European Journal of Combinatorics, 28, 2007, 17241741.


LINKS

Table of n, a(n) for n=0..58.


FORMULA

G.f. = G(t,z)=(1+z)(1z)^2/[1(2+t)z2z^2+2z^3].
G.f. for column k is z^k*(1+z)(1z)^2/(12z2z^2+2z^3)^{k+1} (we have a Riordan array).
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)(1z)^2/[(1+z)(1z)^2(t+s)zsz^2*(1z)].


EXAMPLE

T(2,1)=3 because we have (0/2), (1,0/0,1), and (0,1/1,0) (the 2compositions are written as (top row / bottom row).
Alternatively, T(2,1)=3 because we have (1/1), (1,0/0,1), and (0,1/1,0) (the 2compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
3,3,1;
7,11,5,1;
18,33,23,7,1;
44,100,87,39,9,1;


MAPLE

G := (1+z)*(1z)^2/(1(2+t)*z2*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


CROSSREFS

Cf. A003480, A181305, A181306, A181336
Sequence in context: A106479 A114422 A127501 * A118408 A079268 A102316
Adjacent sequences: A181301 A181302 A181303 * A181305 A181306 A181307


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Oct 13 2010


EXTENSIONS

Edited by N. J. A. Sloane, Oct 15 2010


STATUS

approved



