[seqfan] Re: T(n,k) guess the formula

[Robert Israel]

In such an array, let x_j be the position of the lowest 1 in column j
(0 if that column has no 1's).  Then we require x_{j+1} <= x_j + 1.
So T(n,k) is the number of sequences of integers (x_1,...,x_k) with
all 0 <= x_i <= n and x_{j+1} <= x_j + 1.

Let M be the (n+1) x (n+1) matrix with entries m_{ij} = 1 for j <= i+1, 
0 otherwise, u the row vector of n 1's, v the column vector consisting 
of 1 followed by n 0's.  Then T(n,k) = u M^k v.  So the ogf for row n is 
G_n(z) = sum_{k=0}^infty T(n,k) z^k = u (I - z M)^(-1) v, which is a 
rational function whose poles are reciprocals of the nonzero eigenvalues 
of M.  The first few are as follows, according to Maple:


                                         1
                          G[1](z) = - --------
                                      -1 + 2 z


                                        1
                         G[2](z) = ------------
                                              2
                                   1 - 3 z + z


                                        1
                      G[3](z) = ------------------
                                (3 z - 1) (-1 + z)


                                         1
                    G[4](z) = - --------------------
                                              2    3
                                -1 + 5 z - 6 z  + z


                                         1
                G[5](z) = - ---------------------------
                                           2
                            (-1 + 2 z) (2 z  - 4 z + 1)


                                        1
                G[6](z) = ------------------------------
                                     3      2
                          (-1 + z) (z  - 9 z  + 6 z - 1)


                                        1
               G[7](z) = -------------------------------
                                     2      2
                         (1 - 3 z + z ) (5 z  - 5 z + 1)


                                         1
           G[8](z) = - -------------------------------------
                                      2       3       4    5
                       -1 + 9 z - 28 z  + 35 z  - 15 z  + z


                                         1
        G[9](z) = - --------------------------------------------
                                           2
                    (3 z - 1) (-1 + 2 z) (z  - 4 z + 1) (-1 + z)


                                         1
        G[10](z) = ---------------------------------------------
                                  2       3       4       5    6
                   1 - 11 z + 45 z  - 84 z  + 70 z  - 21 z  + z

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada

On Mon, 4 Apr 2011, Ron Hardin wrote:

> Can the entire formula be guessed?
>
> T(n,k)=Number of nXk binary arrays without the pattern 0 1 diagonally or
> vertically
>
> Table starts
> ..2..4...8...16...32....64....128....256.....512.....1024.....2048......4096
> ..3..8..21...55..144...377....987...2584....6765....17711....46368....121393
> ..4.13..40..121..364..1093...3280...9841...29524....88573...265720....797161
> ..5.19..66..221..728..2380...7753..25213...81927...266110...864201...2806272
> ..6.26.100..364.1288..4488..15504..53296..182688...625184..2137408...7303360
> ..7.34.143..560.2108..7752..28101.100947..360526..1282735..4552624..16131656
> ..8.43.196..820.3264.12597..47652.177859..657800..2417416..8844448..32256553
> ..9.53.260.1156.4845.19551..76912.297275.1134705..4292145.16128061..60304951
> .10.64.336.1581.6954.29260.119416.476905.1874730..7283640.28048800.107286661
> .11.76.425.2109.9709.42504.179630.740025.2991495.11920740.46981740.183579396
>
> Empirical recurrences for rows:
> T(n,k) = sum( binomial(n+2-i ,i) * T(n,k-i) * (-1)^(i-1) , i=1..floor((n+2)/2) )
>
> e.g., a(n) for rows 1..8
> Empirical: a(n)=2*a(n-1)
> Empirical: a(n)=3*a(n-1)-a(n-2)
> Empirical: a(n)=4*a(n-1)-3*a(n-2)
> Empirical: a(n)=5*a(n-1)-6*a(n-2)+a(n-3)
> Empirical: a(n)=6*a(n-1)-10*a(n-2)+4*a(n-3)
> Empirical: a(n)=7*a(n-1)-15*a(n-2)+10*a(n-3)-a(n-4)
> Empirical: a(n)=8*a(n-1)-21*a(n-2)+20*a(n-3)-5*a(n-4)
> Empirical: a(n)=9*a(n-1)-28*a(n-2)+35*a(n-3)-15*a(n-4)+a(n-5)
>
> Empirical polynomials for columns:
> T(n,1) = n + 1
> T(n,2) = (1/2)*n^2 + (5/2)*n + 1
> T(n,3) = (1/6)*n^3 + 2*n^2 + (35/6)*n
> T(n,4) = (1/24)*n^4 + (11/12)*n^3 + (155/24)*n^2 + (163/12)*n - 6 for n>1
> T(n,5) = (1/120)*n^5 + (7/24)*n^4 + (89/24)*n^3 + (473/24)*n^2 + (1877/60)*n -
> 33 for n>2
> T(n,6) = (1/720)*n^6 + (17/240)*n^5 + (203/144)*n^4 + (647/48)*n^3 +
> (2659/45)*n^2 + (1379/20)*n - 143 for n>3
> T(n,7) = (1/5040)*n^7 + (1/72)*n^6 + (143/360)*n^5 + (53/9)*n^4 +
> (33667/720)*n^3 + (12679/72)*n^2 + (9439/70)*n - 572 for n>4
> T(n,8) = (1/40320)*n^8 + (23/10080)*n^7 + (17/192)*n^6 + (269/144)*n^5 +
> (43949/1920)*n^4 + (228401/1440)*n^3 + (1054411/2016)*n^2 + (9941/56)*n - 2210
> for n>5
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
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