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>