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A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries (0<=k<=n) 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. 3
1, 0, 2, 1, 0, 6, 0, 8, 0, 16, 3, 0, 35, 0, 44, 0, 28, 0, 132, 0, 120, 8, 0, 160, 0, 460, 0, 328, 0, 92, 0, 748, 0, 1528, 0, 896, 21, 0, 642, 0, 3117, 0, 4916, 0, 2448, 0, 290, 0, 3552, 0, 12062, 0, 15456, 0, 6688, 55, 0, 2380, 0, 17119, 0, 44318, 0, 47760, 0, 18272, 0, 888, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sum of entries in row n is A003480(n).

T(2n-1,0)=0.

T(2n,0)=A000045(2n) (Fibonacci numbers).

T(n,k)=0 if n and k have opposite parities.

T(n,n)=A002605(n+1).

Sum(k*T(n,k),k=0..n)=A181298.

For the statistics "number of odd entries" see A181295.

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..68.

FORMULA

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

The g.f. H(t,s,z), where z marks the size of the 2-composition and t (s) marks the number of odd (even) entries, is H=1/(1-h), where h=z(t+sz)(2s+tz-sz^2)/(1-z^2)^2.

EXAMPLE

T(2,2)=6 because we have (0 / 2), (2 / 0), (1,0 / 0,1), (0,1 / 1,0), (1,1 / 0,0), (0,0 / 1,1) (the 2-compositions are written as (top row / bottom row).

Triangle starts:

1;

0,2;

1,0,6;

0,8,0,16;

3,0,35,0,44;

MAPLE

G := (1-z^2)^2/(1-3*z^2+z^4-2*s*z-2*s^2*z^2+s^2*z^4): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], s, k), k = 0 .. n) end do; # yields sequence in triangular form

CROSSREFS

Cf. A003480, A000045, A181295, A181296, A181298.

Sequence in context: A266904 A299198 A137477 * A196776 A157982 A119275

Adjacent sequences:  A181294 A181295 A181296 * A181298 A181299 A181300

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 12 2010

STATUS

approved

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Last modified September 18 04:44 EDT 2020. Contains 337165 sequences. (Running on oeis4.)