# [seqfan] Re: Oops!^2 final try, guess the formula http://oeis.org/A200785

Max Alekseyev maxale at gmail.com
Wed Nov 23 00:01:10 CET 2011

The definition of A200785 was somewhat obscure - I've edited it.
It is also not clear why it refers to arrays of length n+2 not just n
- that would be more general sequence with two additional rows:
n=1: T(1,k) = (k+1)
n=2: T(2,k) = (k+1)^2

The inclusion-exclusion implies the following general formula (here I
refer to arrays of length n not n+2):

T(n,k) = \sum_{L=0}^n (-1)^L / L! * \sum_{M=0}^{min(L,[(n-L)/2])}
binomial(n-L-M,M) * M! * (k+1)^(n-L-2*M) B_{L,M}( x_1, x_2, ... )

where B_{L,M}(  x_1, x_2, ... ) are Bell polynomials whose arguments
are given by the formula:

x_i = binomial(k+1,i+2) * i! * f(i),   i=1,2,...
and
f(i) has period of length 6: [0,1,1,0,-1,-1] (i.e., f(0)=0, f(1)=1, etc.).

The general formula implies that for a fixed n, T(n,k) is a polynomial
in k (or better say in (k+1) since it is more natural for this
problem).
This polynomial is easy to compute.

Regards,
Max

On Tue, Nov 22, 2011 at 11:13 PM, Ron Hardin <rhhardin at att.net> wrote:
> (wysinwyg web mail editor is perverse and impossible, starting fresh)
>
> It might be possible to guess the formula for T(n,k) in http://oeis.org/A200785
> from the formula for column 2 in http://oeis.org/A076264
>
> a(n)=sum{k=0..floor(n/3), binomial(n-2k, k)(-1)^k*3^(n-3k)}
>
> with the right generalization.
>
>
>  rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
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