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A062204 Number of alignments of n strings of length 7. 4
1, 1, 48639, 75494983297, 1177359342144641535, 103746115308050354021387521, 36585008462723983824862891403150079, 41020870889694863957061607086939138327565057, 124069835911824710311393852646151897334844371419287295 (list; graph; refs; listen; history; text; internal format)
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^(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
REFERENCES
M. S. Waterman, Introduction to Computational Biology: Maps, Sequences and Genomes, 1995.
LINKS
M. A. Covington, The number of distinct alignments of two strings, Journal of Quantitative Linguistics, Volume 11, no. 3 (2004), 173-182.
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(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
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 cross-validated 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
Cf. A062205, A062208, A001850. A(2, X) represents Waterman's f function.
Row n=7 of A262809.
Sequence in context: A244172 A245794 A048341 * A173780 A252444 A237146
KEYWORD
nonn
AUTHOR
Angelo Dalli, Jun 13 2001
EXTENSIONS
Formula and sequence revised by Max Alekseyev, Mar 12 2009
STATUS
approved

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Last modified June 15 01:02 EDT 2024. Contains 373402 sequences. (Running on oeis4.)