

A181330


Triangle read by rows: T(n,k) is the number of 2compositions of n having k 0's 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.


2



1, 1, 1, 3, 3, 1, 8, 10, 5, 1, 21, 32, 21, 7, 1, 55, 99, 80, 36, 9, 1, 144, 299, 286, 160, 55, 11, 1, 377, 887, 978, 650, 280, 78, 13, 1, 987, 2595, 3236, 2482, 1275, 448, 105, 15, 1, 2584, 7508, 10438, 9054, 5377, 2261, 672, 136, 17, 1, 6765, 21526, 32991, 31882
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

The sum of entries in row n is A003480(n).
T(n,0) = A000045(2n) (n>=1), Fibonacci numbers.
T(n,1) = A038731(n1) (n>=1).
Sum(k*T(n,k), k>=0) = A181331.
For the statistic "number of nonzero entries in the top row" see A181332.


LINKS

Table of n, a(n) for n=0..58.
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of Lconvex polyominoes, European J. Combin. 28 (2007), no. 6, 17241741.


FORMULA

G.f.: G(t,x) = (1x)^2/(13*x+x^2t*x(1x)).
The g.f. of column k is x^k*(1x)^(k+2)/(13*x+x^2)^(k+1) (we have a Riordan array).
T(n,k) = 3*T(n1,k) +T(n1,k1) T(n2,k) T(n2,k1), with T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=T(2,1)=3, T(n,k)=0 if k<0 or if k>n.  Philippe Deléham, Nov 26 2013


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)).
Triangle starts:
1;
1,1;
3,3,1;
8,10,5,1;
21,32,21,7,1;
55,99,80,36,9,1;


MAPLE

G := (1z)^2/(13*z+z^2t*z*(1z)): 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, A000045, A038731, A181331, A181332.
Sequence in context: A160332 A293294 A200342 * A262143 A284554 A078033
Adjacent sequences: A181327 A181328 A181329 * A181331 A181332 A181333


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Oct 13 2010


STATUS

approved



