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A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even 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. 2
1, 1, 1, 2, 4, 1, 5, 11, 7, 1, 11, 31, 29, 10, 1, 25, 83, 102, 56, 13, 1, 56, 217, 329, 245, 92, 16, 1, 126, 556, 1000, 938, 487, 137, 19, 1, 283, 1403, 2917, 3292, 2180, 855, 191, 22, 1, 636, 3498, 8247, 10865, 8740, 4406, 1376, 254, 25, 1, 1429, 8636, 22756, 34248 (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)=A006054(n+1) (n>=1).

Sum(k*T(n,k), k>=0)=A181337(n).

For the statistic "number of odd entries in the top row" see A181304.

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..58.

FORMULA

G.f.=G(s,z)=(1+z)(1-z)^2/[1-2z-z^2+z^3-sz(1+z-z^2)].

The g.f. of column k is z^k*(1+z)(1-z)^2*(1+z-z^2)^k/(1-2z-z^2+z^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)(1-z)^2/[(1+z)(1-z)^2-(t+s)z-sz^2*(1-z)].

EXAMPLE

T(2,1)=4 because we have (0/2), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).

Triangle starts:

1;

1,1;

2,4,1;

5,11,7,1;

11,31,29,10,1;

25,83,102,56,13,1;

MAPLE

G := (1+z)*(1-z)^2/(1-2*z-z^2+z^3-s*z*(1+z-z^2)): Gser := simplify(series(G, z = 0, 13)): 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], s, k), k = 0 .. n) end do; # yields sequence in triangular form

CROSSREFS

Cf. A003480, A006054, A181304, A181337,

Sequence in context: A080427 A118906 A085059 * A238731 A124037 A126126

Adjacent sequences:  A181333 A181334 A181335 * A181337 A181338 A181339

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 14 2010

STATUS

approved

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Last modified October 23 14:43 EDT 2019. Contains 328345 sequences. (Running on oeis4.)