login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A062208 a(n) = Sum_{m>=0} binomial(m,3)^n*2^(-m-1). 7
1, 1, 63, 16081, 10681263, 14638956721, 35941784497263, 143743469278461361, 874531783382503604463, 7687300579969605991710001, 93777824804632275267836362863, 1537173608464960118370398000894641, 32970915649974341628739088902163732463 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Number of alignments of n strings of length 3.
Conjectures: a(2*n) = 3 (mod 60) and a(2*n+1) = 1 (mod 60); for fixed k, the sequence a(n) (mod k) eventually becomes periodic with exact period a divisor of phi(k), where phi(k) is Euler's totient function A000010. - Peter Bala, Feb 04 2018
LINKS
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522
FORMULA
From Vaclav Kotesovec, Mar 22 2016: (Start)
a(n) ~ 3^(2*n + 1/2) * n!^3 / (Pi * n * 2^(n+3) * (log(2))^(3*n+1)).
a(n) ~ sqrt(Pi)*3^(2*n+1/2)*n^(3*n+1/2) / (2^(n+3/2)*exp(3*n)*(log(2))^(3*n+1)).
(End)
a(n) = Sum_{k = 3..3*n} Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)* binomial(i,3)^n. Row sums of A299041. - Peter Bala, Feb 04 2018
MAPLE
A000629 := proc(n) local k ; sum( k^n/2^k, k=0..infinity) ; end: A062208 := proc(n) local a, stir, ni, n1, n2, n3, stir2, i, j, tmp ; a := 0 ; if n = 0 then RETURN(1) ; fi ; stir := combinat[partition](n) ; stir2 := {} ; for i in stir do if nops(i) <= 3 then tmp := i ; while nops(tmp) < 3 do tmp := [op(tmp), 0] ; od: tmp := combinat[permute](tmp) ; for j in tmp do stir2 := stir2 union { j } ; od: fi ; od: for ni in stir2 do n1 := op(1, ni) ; n2 := op(2, ni) ; n3 := op(3, ni) ; a := a+combinat[multinomial](n, n1, n2, n3)*(A000629(3*n1+2*n2+n3)-1/2-2^(3*n1+2*n2+n3)/4)*(-3)^n2*2^n3 ; od: a/(2*6^n) ; end: seq(A062208(n), n=0..14) ; # R. J. Mathar, Apr 01 2008
a:=proc(n) options operator, arrow: sum(binomial(m, 3)^n*2^(-m-1), m=0.. infinity) end proc: seq(a(n), n=0..12); # Emeric Deutsch, Mar 22 2008
MATHEMATICA
a[n_] = Sum[2^(-1-m)*((m-2)*(m-1)*m)^n, {m, 0, Infinity}]/6^n; a /@ Range[0, 12] (* Jean-François Alcover, Jul 13 2011 *)
With[{r = 3}, Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 15}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)
CROSSREFS
See A062204 for further references, formulas and comments.
Row n=3 of A262809.
Sequence in context: A234629 A270507 A289875 * A132594 A212932 A349902
KEYWORD
nonn,easy
AUTHOR
Angelo Dalli, Jun 13 2001
EXTENSIONS
New definition from Vladeta Jovovic, Mar 01 2008
Edited by N. J. A. Sloane, Sep 19 2009 at the suggestion of Max Alekseyev
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 02:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)