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A062204
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Number of alignments of n strings of length 7.
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4
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1, 1, 48639, 75494983297, 1177359342144641535, 103746115308050354021387521, 36585008462723983824862891403150079, 41020870889694863957061607086939138327565057, 124069835911824710311393852646151897334844371419287295
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OFFSET
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0,3
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COMMENTS
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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^(r-1)/(r-1)!)^n * n^(r*n+1/2) / (2^(r/2) * exp(r*n) * (log(2))^(r*n+1)). - Vaclav Kotesovec, Mar 23 2016
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REFERENCES
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M. S. Waterman, Introduction to Computational Biology: Maps, Sequences and Genomes, 1995.
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LINKS
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Michael S. Waterman, Home Page (contains copies of his papers)
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FORMULA
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A(n, y) = sum(k=0,n*y, sum(t=0,k, (-1)^t * binomial(k,t) * binomial(k-t,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
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EXAMPLE
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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 cross-validated against the Delannoy sequence D(X,X) A001850.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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