

A062204


Number of alignments of n strings of length 7.


4



1, 1, 48639, 75494983297, 1177359342144641535, 103746115308050354021387521, 36585008462723983824862891403150079, 41020870889694863957061607086939138327565057, 124069835911824710311393852646151897334844371419287295
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OFFSET

0,3


COMMENTS

Strings of length 7 represent the average word length for most natural languages such as English. This sequence represents the search space for alignment and sequencing algorithms that work on multiple sets of strings.
The assertion that "strings of length 7 represent the average word length for most natural languages such as English" seems to conflict with studies that show that the average word length in English is about 4.5 letters and the average word length in modern Russian is 5.28 letters.  M. F. Hasler, Mar 12 2009
In general, row r > 0 of A262809 is asymptotic to sqrt(r*Pi) * (r^(r1)/(r1)!)^n * n^(r*n+1/2) / (2^(r/2) * exp(r*n) * (log(2))^(r*n+1)).  Vaclav Kotesovec, Mar 23 2016


REFERENCES

M. S. Waterman, Introduction to Computational Biology: Maps, Sequences and Genomes, 1995.


LINKS

Michael S. Waterman, Home Page (contains copies of his papers)


FORMULA

A(n, y) = sum(k=0,n*y, sum(t=0,k, (1)^t * binomial(k,t) * binomial(kt,y)^n )).
a(n) ~ sqrt(7*Pi) * (7^6/6!)^n * n^(7*n+1/2) / (2^(7/2) * exp(7*n) * (log(2))^(7*n+1)).  Vaclav Kotesovec, Mar 23 2016


EXAMPLE

A(2, 7) = 48639 since this represents the number of distinct alignments of 2 strings of length 7. All values in A(2,X) can be crossvalidated against the Delannoy sequence D(X,X) A001850.


MATHEMATICA

With[{r = 7}, Flatten[{1, Table[Sum[Sum[(1)^i*Binomial[j, i]*Binomial[j  i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 10}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



