A109960 -id:A109960 - cp's OEIS frontend

A109962 Inverse of Riordan array (1/(1-x), x/(1-x)^4), A109960.

1, -1, 1, 4, -5, 1, -22, 30, -9, 1, 140, -200, 72, -13, 1, -969, 1425, -570, 130, -17, 1, 7084, -10626, 4554, -1196, 204, -21, 1, -53820, 81900, -36855, 10647, -2142, 294, -25, 1, 420732, -647280, 302064, -93496, 21080, -3472, 400, -29, 1, -3362260, 5217300, -2504304, 816816, -200277, 37485, -5250, 522

Offset: 0

Paul Barry, Jul 06 2005

Riordan array (g,f) where f/(1-f)^4=x and g=1/(1-f). First column is (-1)^n*A002293(n). Diagonal sums are A109963.

			Triangle begins:
     1;
    -1,    1;
     4,   -5,    1;
   -22,   30,   -9,   1;
   140, -200,   72, -13,   1;
  -969, 1425, -570, 130, -17, 1;
  ...

Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 14.
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8 (up to signs).

Cf. A002293, A109960, A109963.

Number triangle T(n, k)=(-1)^(n-k)*((4k+1)/(3n+k+1))*binomial(4n, n-k).

A055988 Sequence is its own 4th difference.

Original entry on oeis.org

1, 2, 7, 26, 95, 345, 1252, 4544, 16493, 59864, 217286, 788674, 2862617, 10390321, 37713313, 136886433, 496850954, 1803399103, 6545722210, 23758733815, 86236081273, 313007493212, 1136110191472, 4123691589365, 14967590689568

Offset: 1

Henry Bottomley, Jun 02 2000

Row sums of Riordan array (1/(1-x), x/(1-x)^4), A109960. - Paul Barry, Jul 06 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).

Cf. A055989, A055990, A055991 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

I:=[1, 2, 7, 26]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 05 2012

CoefficientList[Series[(1-x)^3/(1-5x+6x^2-4x^3+x^4),{x,0,40}],x] (* Vincenzo Librandi, Apr 05 2012 *)
LinearRecurrence[{5,-6,4,-1},{1,2,7,26},30] (* Harvey P. Dale, Jan 15 2017 *)

a(n) = 5a(n-1) - 6a(n-2) + 4a(n-3) - a(n-4) = a(n-1) + A055991(n-1) = A055989(n) - A055989(n-1) = A055990(n) - 2*A055990(n-1) + A055990(n-2).
From Paul Barry, Jul 06 2005: (Start)
G.f.: (1-x)^3/(1 - 5x + 6x^2 - 4x^3 + x^4);
a(n) = Sum_{k=0..n} binomial(n+3k, 4k). (End)

More terms from James Sellers, Jun 05 2000

A109961 Expansion of (1-x)^3/(1-4x+5x^2-4x^3+x^4).

Original entry on oeis.org

1, 1, 2, 6, 17, 45, 117, 305, 798, 2090, 5473, 14329, 37513, 98209, 257114, 673134, 1762289, 4613733, 12078909, 31622993, 82790070, 216747218, 567451585, 1485607537, 3889371025, 10182505537, 26658145586, 69791931222, 182717648081

Offset: 0

Paul Barry, Jul 06 2005

Diagonal sums of number triangle A109960.

Index entries for linear recurrences with constant coefficients, signature (4,-5,4,-1).

CoefficientList[Series[(1-3x+3x^2-x^3)/(1-4x+5x^2-4x^3+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{4,-5,4,-1},{1,1,2,6},40] (* Harvey P. Dale, Dec 11 2013 *)

a(n)=sum{k=0..floor(n/2), binomial(n+2k, 4k)}.

a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(n)=4*a(n-1)-5*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Dec 11 2013

A236579 The number of tilings of a 5 X (4n) floor with 1 X 4 tetrominoes.

Original entry on oeis.org

1, 3, 15, 75, 371, 1833, 9057, 44753, 221137, 1092699, 5399327, 26679563, 131831075, 651413681, 3218814561, 15905050017, 78591236385, 388340962771, 1918899743823, 9481812581835, 46852249642771

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

Related to A002378 by an Invert Transform.

Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 34.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], eq. (24).
Index entries for linear recurrences with constant coefficients, signature (6,-6,4,-1).

Cf. A003269 (4Xn floor), A236580 - A236582, A109960.

g := (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4) ;
taylor(%,x=0,30) ; gfun[seriestolist](%) ;
# Alternatively:
a := n -> hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128):
seq(simplify(a(n)), n=0..20); # Peter Luschny, Nov 02 2017

LinearRecurrence[{6, -6, 4, -1}, {1, 3, 15, 75}, 21] (* Jean-François Alcover, Jul 14 2018 *)

G.f.: (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4).
a(n) = Sum_{k = 0..n} binomial(n + 3*k, 4*k)*2^k = Sum_{k = 0..n} A109960(n,k)*2^k. - Peter Bala, Nov 02 2017
a(n) = hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128). - Peter Luschny, Nov 02 2017

cp's OEIS Frontend

A109962 Inverse of Riordan array (1/(1-x), x/(1-x)^4), A109960.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A055988 Sequence is its own 4th difference.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A109961 Expansion of (1-x)^3/(1-4x+5x^2-4x^3+x^4).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A236579 The number of tilings of a 5 X (4n) floor with 1 X 4 tetrominoes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula