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Re: Function fits with combinations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122934] Re: Function fits with combinations
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Thu, 17 Nov 2011 06:04:19 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <ja015e$64t$1@smc.vnet.net>

In such cases I always start by consulting the Sloan page: 
http://oeis.org

And violà, your function on the right hand sides is clearly A027907

The values are the coefficients of the expansion of (1+x+x^2)^n
Clear[x, n];
Table[{n, Expand[(1 + x + x^2)^n]}, {n, 1, 5}];
{1, 1 + x + x^2}
{2, 1 + 2*x + 3*x^2 + 2*x^3 + x^4}
{3, 1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6}
{4, 1 + 4*x + 10*x^2 + 16*x^3 + 19*x^4 + 16*x^5 + 10*x^6 + 4*x^7 + x^8}
{5, 1 + 5*x + 15*x^2 + 30*x^3 + 45*x^4 + 51*x^5 + 45*x^6 + 30*x^7 + 
15*x^8 + 5*x^9 + x^10}

An explicit formula is (cf. link above)

Table[Sum[Binomial[n, i]*Binomial[n - i, k - 2*i], {i, 0, n}], {n, 0, 
5}, {k, 0, 2*n}];
{1}
{1, 1, 1}
{1, 2, 3, 2, 1}
{1, 3, 6, 7, 6, 3, 1}
{1, 4, 10, 16, 19, 16, 10, 4, 1}
{1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1}

Best ragards,
Wolfgang


"richardmathur" <rickarrano at gmail.com> schrieb im Newsbeitrag 
news:ja015e$64t$1 at smc.vnet.net...
> Hello,
> I've been attempting to use Wolfram to help me identify a function 
> but
> after playing with "fit" it keeps giving me linear/quadratic/etc.
> solutions and I am sure that the function generating my data is using
> combinations. I have a series of n length/m length pairs and I've 
> been
> generating alignments between the two, and in the first segment of 
> the
> pair any character can map to 1-3 chars of the latter. The number of
> alignments is as follows:
>
> 3,3 = 1
> 3,4 = 3
> 3,5 = 6
> 3,6 = 7
> 3,7 = 6
> 3,8 = 3
> 3,9 = 1
>
> 4,4 = 1
> 4,5 = 4
> 4,6 = 10
> 4,7 = 16
> 4,8 = 19
> 4,9 = 16
> 4,10 = 10
> 4,11 = 4
> 4,12 = 1
>
> 5,5 = 1
> 5,6 = 5
> 5,7 = 15
> 5,8 = 30
> 5,9 = 45
> 5,10 = 51
> 5,11 = 45
> 5,12 = 30
> 5,13 = 15
> 5,14 = 5
> 5,15 = 1
>
> Can anyone help me to either identify the function in question or
> figure out how to point to Wolfram that it almost certainly has to do
> with combinations?
>
> Thanks,
> Richard
> 




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