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A181307 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with only nonzero entries (0<=k<=floor(n/2)). 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. 0
1, 2, 6, 1, 18, 6, 54, 27, 1, 162, 108, 10, 486, 405, 64, 1, 1458, 1458, 334, 14, 4374, 5103, 1549, 117, 1, 13122, 17496, 6652, 760, 18, 39366, 59049, 27064, 4238, 186, 1, 118098, 196830, 105796, 21324, 1450, 22, 354294, 649539, 401041, 99646, 9480, 271 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of entries in row n is 1+floor(n/2).

Sum of entries in row n is A003480(n).

T(n,0)=2*3^{n-1}=A008776(n-1).

Sum(k*T(n,k),k>=0)=A054146(n-1).

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..47.

FORMULA

G.f. = G(t,z)=(1-z)^2/(1-4z+3z^2-tz^2).

The g.f. of column k is z^{2k}/[(1-3z)^{k+1}*(1-z)^{k-1}] (we have a Riordan array).

EXAMPLE

T(2,1)=1 because we have (1/1) (the 2-compositions are written as (top row / bottom row).

Triangle starts:

1;

2;

6,1;

18,6;

54,27,1;

162,108,10;

MAPLE

G := (1-z)^2/(1-4*z+3*z^2-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

CROSSREFS

Cf. A003480, A008776, A054146.

Sequence in context: A281307 A281417 A175353 * A008855 A181299 A181365

Adjacent sequences:  A181304 A181305 A181306 * A181308 A181309 A181310

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Oct 13 2010

STATUS

approved

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Last modified September 21 23:55 EDT 2019. Contains 327286 sequences. (Running on oeis4.)