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 >

**Follow-Ups**:**Re: Function fits with combinations***From:*DrMajorBob <btreat1@austin.rr.com>