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A181308 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an odd sum (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, 3, 0, 4, 0, 16, 0, 8, 14, 0, 52, 0, 16, 0, 104, 0, 144, 0, 32, 64, 0, 460, 0, 368, 0, 64, 0, 616, 0, 1624, 0, 896, 0, 128, 292, 0, 3428, 0, 5056, 0, 2112, 0, 256, 0, 3456, 0, 14688, 0, 14528, 0, 4864, 0, 512, 1332, 0, 23132, 0, 53920, 0, 39488, 0, 11008, 0, 1024, 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(n,k) = 0 if n and k have opposite parities.

T(2n,0) = A060801(n).

Sum(k*T(n,k), k=0..n) = A181326(n).

For the statistic "number of column with an even sum" see A181327.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

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,z) = (1-z)^2*(1+z)^2/(1-5z^2+2z^4-2tz).

The g.f. of column k is (2z)^k*(1-z^2)^2/(1-5z^2+2z^4)^{k+1} (we have a Riordan array).

The g.f. H(t,s,z), where z marks size and t (s) marks number of columns with an odd (even) sum, is H=(1-z^2)^2/(1-2z^2+z^4-2tz-3sz^2+sz^4).

EXAMPLE

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

Triangle starts:

1;

0,  2;

3,  0,  4;

0, 16,  0, 8;

14, 0, 52, 0, 16;

MAPLE

G := (1-z^2)^2/(1-5*z^2+2*z^4-2*t*z): 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], t, k), k = 0 .. n) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n) option remember; `if`(n=0, 1,

       expand(add(add(`if`(i=0 and j=0, 0, b(n-i-j)*

       `if`(irem(i+j, 2)=1, x, 1)), i=0..n-j), j=0..n)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):

seq(T(n), n=0..15); # Alois P. Heinz, Mar 16 2014

MATHEMATICA

b[n_] := b[n] = If[n == 0, 1, Expand[Sum[Sum[If[i == 0 && j == 0, 0, b[n-i-j]* If[Mod[i+j, 2] == 1, x, 1]], {i, 0, n-j}], {j, 0, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-Fran├žois Alcover, Feb 19 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A003480, A060801, A181326, A181327.

Sequence in context: A166238 A293275 A014197 * A292246 A277141 A021438

Adjacent sequences:  A181305 A181306 A181307 * A181309 A181310 A181311

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 13 2010

STATUS

approved

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Last modified October 16 06:02 EDT 2019. Contains 328046 sequences. (Running on oeis4.)