A055995 -id:A055995 - cp's OEIS frontend

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1

Offset: 0

Philippe Deléham, Dec 05 2011

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)b(i-1,j)-c(i,j)c(i,j-1) is nonzero.

Original entry on oeis.org

36, 144, 144, 576, 864, 576, 2304, 5184, 5184, 2304, 9216, 31104, 46656, 31104, 9216, 36864, 186624, 419904, 419904, 186624, 36864, 147456, 1119744, 3779136, 5738688, 3779136, 1119744, 147456, 589824, 6718464, 34012224, 78428736, 78428736

Offset: 1

R. H. Hardin Jan 10 2012

Also 0..2 arrays with no 2X2 subblock having equal diagonal elements or equal antidiagonal elements
Table starts
.....36......144........576.........2304...........9216............36864
....144......864.......5184........31104.........186624..........1119744
....576.....5184......46656.......419904........3779136.........34012224
...2304....31104.....419904......5738688.......78428736.......1073134656
...9216...186624....3779136.....78428736.....1631513664......34026967296
..36864..1119744...34012224...1073134656....34026967296....1084257353088
.147456..6718464..306110016..14683622976...710001723456...34589078037504
.589824.40310784.2754990144.200937920832.14819050600704.1104253773912576

			Some solutions for n=5 k=3
..0..1..0..1....1..2..0..1....0..1..2..1....2..2..0..1....2..2..2..1
..2..1..2..1....1..2..0..1....2..1..0..0....0..1..0..2....0..0..0..1
..0..0..0..1....1..2..0..2....2..1..2..1....2..1..0..1....2..2..2..2
..1..1..2..1....1..2..0..1....2..0..2..0....2..1..2..2....0..0..0..1
..0..0..2..0....1..2..0..1....1..0..2..0....0..0..0..0....2..2..2..2
..1..1..1..1....0..2..0..1....1..0..1..1....1..2..1..2....1..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..285

Column 1 is A002063
Column 2 is A067411(n+2)
Column 3 is A055995(n+2)

Empirical for column k:
k=1: T(n,k)=4*T(n-1,k)
k=2: T(n,k)=6*T(n-1,k)
k=3: T(n,k)=9*T(n-1,k)
k=4: T(n,k)=15*T(n-1,k)-270*T(n-3,k)+324*T(n-4,k)
k=5: T(n,k)=25*T(n-1,k)-45*T(n-2,k)-963*T(n-3,k)+2025*T(n-4,k)+3645*T(n-5,k)-6561*T(n-6,k)
k=6: (order 15)
k=7: (order 45)

cp's OEIS Frontend

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)b(i-1,j)-c(i,j)c(i,j-1) is nonzero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

cp's OEIS Frontend

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)*b(i-1,j)-c(i,j)*c(i,j-1) is nonzero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)b(i-1,j)-c(i,j)c(i,j-1) is nonzero.