cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

Emeric Deutsch, Oct 12 2010

Keywords

Comments

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.

Examples

			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;

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.

Crossrefs

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

Programs

  • Maple
    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

Formula

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

Views

Author

Emeric Deutsch, Oct 12 2010

Keywords

Comments

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.

Examples

			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;

Links

Crossrefs

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

Programs

  • 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

Formula

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).

A181298 The number of even entries in all the 2-compositions of 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

0, 2, 12, 56, 246, 1024, 4128, 16248, 62832, 239640, 903944, 3379064, 12536552, 46215672, 169443592, 618303864, 2246863624, 8135066488, 29358346888, 105642047864, 379143054472, 1357496762744, 4849952390792, 17293404551544
Offset: 0

Views

Author

Emeric Deutsch, Oct 12 2010

Keywords

Comments

a(n)=Sum(k*A181297(n,k),k=0..n).

Examples

			a(2)=12 because in the 2-compositions of 2, namely (1/1),(0/2),(2/0),(1,0/0,1),(0,1/1,0),(1,1/0,0), and (0,0/1,1), we have 0+2+2+2+2+2+2=12 odd entries (the 2-compositions are written as (top row/bottom row)).

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

Crossrefs

Cf. A181295, A181296, A181297.

Programs

  • Maple
    g := 2*z*(1-z)^2*(1+z-z^2)/((1+z)*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 25);

Formula

G.f. = 2z(1-z)^2*(1+z-z^2)/[(1+z)(1-4z+2z^2)^2].
a(n) = 2*A181337(n). - R. J. Mathar, Jul 26 2022

A181302 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with distinct entries (0<=k<=n).

Original entry on oeis.org

1, 0, 2, 1, 2, 4, 0, 8, 8, 8, 2, 8, 32, 24, 16, 0, 24, 56, 104, 64, 32, 4, 24, 152, 248, 304, 160, 64, 0, 64, 248, 712, 896, 832, 384, 128, 8, 64, 568, 1496, 2800, 2880, 2176, 896, 256, 0, 160, 888, 3560, 6976, 9824, 8576, 5504, 2048, 512, 16, 160, 1848, 6904, 17904
Offset: 0

Views

Author

Emeric Deutsch, Oct 13 2010

Keywords

Comments

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.

Examples

			T(2,1) = 2 because we have (0/2) and (2/0) (the 2-compositions are written as (top row/bottom row)).
Triangle starts:
  1;
  0,2;
  1,2,4;
  0,8,8,8;
  2,8,32,24,16;

Links

Crossrefs

Cf. A003480, A181296.

Programs

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

Formula

G.f.: G(t,z) = (1+z)*(1-z)^2/((1-z)*(1-2*z^2)-2*t*z).
G.f. of column k: 2^k*z^k*(1+z)/((1-2*z^2)^(k+1)*(1-z)^(k-1)) (we have a Riordan array).
Sum_{k>=0} k*T(n,k) = A181296(n).
Sum_{k>=0} T(n,k) = A003480(n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.