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 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 Alois P. Heinz, Table of n, a(n) for n = 0..100 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 Cf. A000670, A055203, A001850, A126086. See A062204 for further references, formulas and comments. Cf. A001850, A062204, A062205, A299041. Row n=3 of A262809. Sequence in context: A234629 A270507 A289875 * A132594 A212932 A177233 Adjacent sequences:  A062205 A062206 A062207 * A062209 A062210 A062211 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

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Last modified April 11 18:59 EDT 2021. Contains 342888 sequences. (Running on oeis4.)