A181305
Number of increasing columns in all 2-compositions of n.
Original entry on oeis.org
0, 1, 5, 24, 104, 432, 1736, 6820, 26332, 100308, 377996, 1411844, 5234428, 19285252, 70670972, 257766212, 936336572, 3388962884, 12226547132, 43983439684, 157814634684, 564917186372, 2017873643708, 7193745818436
Offset: 0
a(1) = 1 because in the 2-compositions of 1, namely (0/1) and (1/0) we have only one increasing column (the 2-compositions are written as (top row / bottom row)).
a(2) = 5 because in (0/2), (1/1), (2,0), (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)) we have 0+1+0+1+1+2+0 = 5 odd entries.
- 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.
- Index entries for linear recurrences with constant coefficients, signature (7,-12,-4,12,-4).
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g := z*(1-z)^2/((1+z)*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, k), k = 0 .. 27);
A181306
Number of 2-compositions of n having no increasing columns.
Original entry on oeis.org
1, 1, 3, 7, 18, 44, 110, 272, 676, 1676, 4160, 10320, 25608, 63536, 157648, 391152, 970528, 2408064, 5974880, 14824832, 36783296, 91266496, 226449920, 561866240, 1394099328, 3459031296, 8582528768, 21294921472, 52836837888, 131098461184
Offset: 0
a(2) = 3 because we have (1/1), (2/0), and (1,1/0,0) (the 2-compositions are written as (top row / bottom row)).
Alternatively, a(2) = 3 because we have (0/2), (2,0), and (0,0/1,1) (the 2-compositions are written as (top row/bottom row)).
- 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.
- Sean A. Irvine, Walks on Graphs.
- Index entries for linear recurrences with constant coefficients, signature (2,2,-2).
-
g := (1+z)*(1-z)^2/(1-2*z-2*z^2+2*z^3): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
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CoefficientList[Series[((1+x)(1-x)^2)/(1-2x-2x^2+2x^3),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{2,2,-2},{1,3,7},30]] (* Harvey P. Dale, Mar 07 2012 *)
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.
Original entry on oeis.org
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
Offset: 0
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;
- 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.
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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
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