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Re: Function fits with combinations

• To: mathgroup at smc.vnet.net
• Subject: [mg122952] Re: Function fits with combinations
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Fri, 18 Nov 2011 06:23:04 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <ja015e$64t$1@smc.vnet.net> <201111171104.GAA22774@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

Sloan's encyclopedia is sometimes a VERY valuable resource!

Bobby

On Thu, 17 Nov 2011 05:04:19 -0600, Dr. Wolfgang Hintze <weh at snafu.de>  
wrote:

> 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
>>
>
>


-- 
DrMajorBob at yahoo.com

• References:
  • Re: Function fits with combinations
    • From: "Dr. Wolfgang Hintze" <weh@snafu.de>