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A181330 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's 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. 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(n-1) (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 L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

FORMULA

G.f.: G(t,x) = (1-x)^2/(1-3*x+x^2-t*x(1-x)).

The g.f. of column k is x^k*(1-x)^(k+2)/(1-3*x+x^2)^(k+1) (we have a Riordan array).

T(n,k) = 3*T(n-1,k) +T(n-1,k-1) -T(n-2,k) -T(n-2,k-1), 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 2-compositions 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 := (1-z)^2/(1-3*z+z^2-t*z*(1-z)): 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

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Last modified April 4 08:58 EDT 2020. Contains 333213 sequences. (Running on oeis4.)