A003480 -id:A003480 - cp's OEIS frontend

A181295 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd 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.

1, 0, 2, 2, 0, 5, 0, 12, 0, 12, 7, 0, 46, 0, 29, 0, 58, 0, 152, 0, 70, 24, 0, 297, 0, 466, 0, 169, 0, 256, 0, 1236, 0, 1364, 0, 408, 82, 0, 1632, 0, 4575, 0, 3870, 0, 985, 0, 1072, 0, 8160, 0, 15702, 0, 10736, 0, 2378, 280, 0, 8160, 0, 35320, 0, 51121, 0, 29282, 0, 5741, 0

Offset: 0

Emeric Deutsch, Oct 12 2010

The sum of entries in row n is A003480(n).
T(2n-1,0)=0.
T(2n,0)=A003480(n).
T(n,k)=0 if n and k have opposite parities.
T(n,n)=A000129(n+1) (the Pell numbers).
Sum(k*T(n,k),k=0..n)=A181296.
For the statistics "number of even entries" see A181297.

			T(2,2)=5 because we have (1/1),(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;
2,0,5;
0,12,0,12;
7,0,46,0,29;

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.

Cf. A003480, A000129, A181296, A181297, A181298,

G := (1-z^2)^2/(1-4*z^2+2*z^4-2*t*z-t^2*z^2): 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

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

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.

A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries (0<=k<=n).

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Oct 12 2010

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.

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

			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;

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.

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

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

G.f.: G(t,z) = (1-z^2)^2/(1-3*z^2+z^4-2*s*z-2*s^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.
Sum_{k=0..n} T(n,k) = A003480(n).
T(2*n-1,0) = 0.
T(2*n,0) = A000045(2*n) (Fibonacci numbers).
T(n,k) = 0 if n and k have opposite parities.
T(n,n) = A002605(n+1).
Sum_{k=0..n} k*T(n,k) = A181298(n).

A181304 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with increasing entries (0<=k<=n).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 7, 11, 5, 1, 18, 33, 23, 7, 1, 44, 100, 87, 39, 9, 1, 110, 288, 310, 177, 59, 11, 1, 272, 820, 1036, 728, 311, 83, 13, 1, 676, 2288, 3338, 2768, 1450, 497, 111, 15, 1, 1676, 6316, 10416, 9976, 6172, 2588, 743, 143, 17, 1, 4160, 17244, 31752, 34448

Offset: 0

Emeric Deutsch, Oct 13 2010

Also, triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd 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.
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.
The sum of entries in row n is A003480(n).
For the statistic "number of even entries in the top row" see A181336.

			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)).
Alternatively, T(2,1) = 3 because we have (1/1), (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;
  7,11,5,1;
  18,33,23,7,1;
  44,100,87,39,9,1;

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.

Cf. A003480, A181305, A181306, A181336.

G := (1+z)*(1-z)^2/(1-(2+t)*z-2*z^2+2*z^3): 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

T(n,0) = A181306(n).
Sum_{k>=0} k*T(n,k) = A181305(n).
G.f.: G(t,z) = (1+z)*(1-z)^2/(1-(2+t)*z-2*z^2+2*z^3).
G.f. for column k: z^k*(1+z)*(1-z)^2/(1-2*z-2*z^2+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)].

Edited by N. J. A. Sloane, Oct 15 2010

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.

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Oct 13 2010

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.

			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;

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.

Cf. A003480, A060801, A181326, A181327.

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

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-François Alcover, Feb 19 2015, after Alois P. Heinz *)

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).

A181365 Triangle read by rows: T(n,k) is the number of 2-compositions of n having least entry equal to k (n >= 1; 0 <= k <= floor(n/2)).

Original entry on oeis.org

2, 6, 1, 22, 2, 78, 3, 1, 272, 6, 2, 940, 13, 2, 1, 3232, 28, 2, 2, 11080, 58, 3, 2, 1, 37920, 118, 6, 2, 2, 129648, 239, 12, 2, 2, 1, 443008, 484, 22, 2, 2, 2, 1513248, 979, 37, 3, 2, 2, 1, 5168000, 1976, 60, 6, 2, 2, 2, 17647552, 3980, 97, 12, 2, 2, 2, 1, 60258304, 8004

Offset: 1

Emeric Deutsch, Oct 15 2010

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.

Row n contains 1 + floor(n/2) entries.

			T(4,1) = 3 because we have (1/3), (3/1), and (1,1/1,1) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
    2;
    6,  1;
   22,  2;
   78,  3, 1;
  272,  6, 2;
  940, 13, 2, 1;

Alois P. Heinz, Rows n = 1..200, 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.

Cf. A003480, A181366, A181367.

h := proc (k) if k = 0 then (1-z)^2/(1-4*z+2*z^2) else (1-z)^2/(1-2*z+z^2-z^(2*k)) end if end proc: f := proc (k) options operator, arrow: h(k)-h(k+1) end proc; G := f(0)+sum(f(k)*t^k, k = 1 .. 30): Gser := simplify(series(G, z = 0, 20)): for n to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
     `if`(i=0 and j=0, 0, A(n-i-j, k)), i=k..n-j), j=k..n))
    end:
T:= (n, k)-> A(n, k) -A(n, k+1):
seq(seq(T(n, k), k=0..n/2), n=1..15); # Alois P. Heinz, Mar 16 2014

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[If[i == 0 && j == 0, 0, A[n-i-j, k]], {i, k, n-j}], {j, k, n}]]; T[n_, k_] := A[n, k] - A[n, k+1]; Table[Table[T[n, k], {k, 0, n/2}], {n, 1, 15}] // Flatten (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

G.f. for 2-compositions with all entries >= k is h(k,z) = (1-z)^2/(1-2*z+z^2-z^(2*k)) if k>0 and h(0,z) = (1-z)^2/(1-4*z+2*z^2) if k=0.
G.f. for 2-compositions with least entry k is f(k,z) = h(k,z)-h(k+1,z) (these are the column g.f.'s).
G.f.: G(t,z) = f(0,z) + Sum_{k>=1} f(k,z)*t^k.
Sum_{k >= 0} T(n,k) = A003480(n).
T(n,1) = A181367(n).
Sum_{k >= 0} k*T(n,k) = A181366(n).

A095934 Expansion of (1-x)^2/(1-5x+3x^2).

Original entry on oeis.org

1, 3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134

Offset: 0

N. J. A. Sloane, Jul 13 2004

a(n) is the number of generalized compositions of n when there are i+2 different types of i, (i=1,2,...). [Milan Janjic, Sep 24 2010]

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A003480, A010903, A010920; equals A095939 + A095940.

CoefficientList[Series[(1-x)^2/(1-5x+3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-3},{1,3,13},30] (* Harvey P. Dale, Jun 21 2021 *)

a(n)=polcoeff((1-x)^2/(1-5*x+3*x^2)+x*O(x^n),n)

a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 1, a(1) = 3, a(2) = 13.

A099880 Number of preferential arrangements (or simple hierarchies) of 2*n labeled elements with two kinds of elements (where each kind has n elements).

Original entry on oeis.org

1, 2, 18, 260, 5250, 136332, 4327092, 162309576, 7024896450, 344582629820, 18890850749628, 1144656941236536, 75963981061424820, 5479642938171428600, 426894499408073653800, 35720957482170932284560, 3195135789350678836128450, 304234032845362459798904220

Offset: 0

Thomas Wieder, Nov 02 2004

nonn

The unlabeled case seems to be given by A003480, which can be generated by the following combstruct command: SeqUnionU := [S, {S=Sequence(Set(U,card>=1), card>=1), U=Union(a,b), a=Atom, b=Atom},unlabeled]; [seq(count(SeqUnionU, size=n), n=0..20)]; .

			Let a[1], a[2],...,a[n] and b[1],b[2],...,b[n] denote two kinds "a" and "b" of labeled elements where each kind as n elements in total.
Let ":" denote a level, e.g., if the elements a[1] and a[2] are on level L=1 and the element b[1] is on level L=2 then a[1]a[2]:b[1] is a preferrential arrangement (a simple hierarchy) with two levels.
Then for n=2 we have a(2) = 18 arrangements: a[1]a[2]; a[1]:a[2]; a[2]:a[1]; a[1]b[1]; a[1]:b[1]; b[1]:a[1]; a[1]b[2]; a[1]:b[2]; b[2]:a[1]; a[2]b[1]; a[2]:b[1]; b[1]:a[2]; a[2]b[2]; a[2]:b[2]; b[2]:a[2]; b[1]b[2]; b[1]:b[2]; b[2]:b[1].

Alois P. Heinz, Table of n, a(n) for n = 0..348

Cf. A000670, A075729, A003480, A154921.

a:=n-> add(binomial(2*n, n)*(Stirling2(n, k))*k!, k=0..n): seq(a(n), n=0..16); # Zerinvary Lajos, Oct 19 2006
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
a:= n-> b(n)*(2*n)!/n!:
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 03 2019

f[n_] := Sum[l! StirlingS2[n, l] Binomial[2n, n], {l, n}]; Table[ f[n], {n, 0, 16}] (* Robert G. Wilson v, Nov 04 2004 *)

a(n) = binomial(2*n, n) * Sum_{k=0..n} k! * Stirling2(n, k).
a(n) = binomial(2*n, n) * A000670(n).
a(n) = A154921(2n,n). - Mats Granvik, Feb 07 2009

More terms from Robert G. Wilson v, Nov 04 2004

a(0) corrected and edited by Alois P. Heinz, Feb 03 2019

A181299 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 2, 6, 1, 20, 4, 64, 17, 1, 206, 68, 6, 662, 261, 32, 1, 2128, 976, 152, 8, 6840, 3577, 675, 51, 1, 21986, 12912, 2860, 280, 10, 70670, 46049, 11704, 1406, 74, 1, 227156, 162628, 46632, 6632, 460, 12, 730152, 569705, 181877, 29866, 2570, 101, 1, 2346942

Offset: 0

Emeric Deutsch, Oct 12 2010

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.

Row n contains 1+floor(n/2) entries.

			T(3,1) = 4 because we have (1,1/1,0), (1,0/1,1), (1,1/0,1), (0,1/1,1) (the 2-compositions are written as (top row/bottom row)).
Triangle starts:
  1;
  2;
  6,1;
  20,4;
  64,17,1;

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.

Cf. A003480, A181300, A181301.

G := (1-z)^2*(1+z)/(1-3*z-z^2+z^3-t*z^2*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1+z)*(1-z)^2/(1-3*z-z^2+z^3-t*(1-z)*z^2).
Sum_{k>=0} T(n,k) = A003480(n).
T(n,0) = A181301(n).
Sum_{k>=0} k*T(n,k) = A181300(n).

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.

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Oct 13 2010

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.

			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;

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.

Cf. A003480, A000045, A038731, A181331, A181332.

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

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

A181332 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k nonzero 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, 4, 12, 7, 1, 8, 32, 31, 10, 1, 16, 80, 111, 59, 13, 1, 32, 192, 351, 268, 96, 16, 1, 64, 448, 1023, 1037, 530, 142, 19, 1, 128, 1024, 2815, 3598, 2435, 924, 197, 22, 1, 256, 2304, 7423, 11535, 9843, 4923, 1477, 261, 25, 1, 512, 5120, 18943, 34832

Offset: 0

Emeric Deutsch, Oct 13 2010

The sum of entries in row n is A003480(n).
T(n,1) = A001787(n).
T(n,2) = A055580(n-2) (n>=2).
T(n,3) = A055586(n-3) (n>=3).
Sum(k*T(n,k), k>=0) = A054146(n).

			T(2,1)=4 because we have (1/1), (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;
4,12,7,1;
8,32,31,10,1;
16,80,111,59,13,1;

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.

Cf. A003480, A001787, A055580, A055586, A181330, A181332.

T := proc (n, k) options operator, arrow: sum(2^j*binomial(k+j, k)*binomial(n-j-2, k-2), j = 0 .. n-k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

T(n,k) = sum(2^j*binomial(k+j,k)*binomial(n-2-j,k-2), j=0..n-k).
G.f.: G(t,x) = (1-x)^2/(1-3*x+2*x^2-t*x).
The g.f. of column k is x^k/((1-2*x)^(k+1)*(1-x)^(k-1)) (we have a Riordan array).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), with T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=2, T(2,1)=4, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham, Nov 26 2013

cp's OEIS Frontend

A181295 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd 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.

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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).

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A181304 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with increasing entries (0<=k<=n).

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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.

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A181365 Triangle read by rows: T(n,k) is the number of 2-compositions of n having least entry equal to k (n >= 1; 0 <= k <= floor(n/2)).

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A095934 Expansion of (1-x)^2/(1-5*x+3*x^2).

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A099880 Number of preferential arrangements (or simple hierarchies) of 2*n labeled elements with two kinds of elements (where each kind has n elements).

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A095934 Expansion of (1-x)^2/(1-5x+3x^2).