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 A124435 Number of effective multiple alignments of three equal-length sequences. 2
 1, 5, 67, 1109, 20251, 391355, 7847155, 161476565, 3387271675, 72114452255, 1553475100717, 33786532319435, 740681494769659, 16346552430326123, 362830907979309067, 8093356178498583509, 181311959402343288955, 4077310062938894133623, 91999289732199733092601 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This counts effective alignments rather than standard alignments, so that for example the following two alignments are equivalent: -A A- -T T- C- -C See Dress, Morgenstern and Stoye for more information. LINKS Robert Israel, Table of n, a(n) for n = 0..720 A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015. A. Dress, B. Morgenstern and J. Stoye, On the number of standard and of effective multiple alignments,  Applied Mathematics Letters, Vol. 11, No. 4, 1998, pp. 43-49. FORMULA The recurrence is three dimensional with the order of the three parameters immaterial. That is, a(i,j,k)=a(i,k,j)=a(j,i,k)=a(j,k,i)=a(k,i,j)=a(k,j,i). a(i, j, 0) = (i+j)! / i! / j! a(i, j, k) = a(i-1,j,k) + a(i,j-1,k) + a(i,j,k-1) - a(i-1,j-1,k-1). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*binomial(n+2*k,n)*binomial(2*k,k). - Wadim Zudilin, Nov 26 2015 Diagonal of 1/(1 - x - y - z + x*y*z). - Mark van Hoeij, Dec 20 2013 G.f.: hypergeom([1/3, 2/3],[1],27*x/(1+x)^3)/(1+x). - Mark van Hoeij, Dec 20 2013 (3*n-1)*(n+1)^2*a(n+1)-(3*n+1)*(24*n^2+8*n-5)*a(n)+(9*n^3-3*n^2-4*n+2)*a(n-1)+(3*n+2)*(n-1)^2*a(n-2)=0. - Robert Israel, Nov 26 2015 0 = (2*x-1)*(x^3+3*x^2-24*x+1)*x*y'' + (6*x^4+8*x^3-57*x^2+48*x-1)*y' + (x+1)*(2*x^2-2*x+5)*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016 EXAMPLE a(1) = 5 because the five alignments are A-- A- A- A- A -C- C- -C -C C --T -T T- -T T MAPLE G := series( hypergeom([1/3, 2/3], [1], 27*x/(1+x)^3)/(1+x), x=0, 31); seq(coeff(G, x, i), i=0..30);  # Mark van Hoeij, Dec 20 2013 MATHEMATICA a[n_] := Sum[(-1)^(n-k) Binomial[n, k] Binomial[n+2k, n] Binomial[2k, k], {k, 0, n}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Sep 18 2018, after Wadim Zudilin *) PROG (PARI) diag(expr, N=22, var=variables(expr)) = {   my(a = vector(N));   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));   for (n = 1, N, a[n] = expr;     for (k = 1, #var, a[n] = polcoef(a[n], n-1)));   return(a); }; x='x; y='y; z='z; diag(1/(1 - x - y - z + x*y*z), 19) (PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi"); read("hypergeom.gpi"); N = 20; x = 'x + O('x^N); Vec(hypergeom([1/3, 2/3], [1], 27*x/(1+x)^3, N)/(1+x)) \\ Gheorghe Coserea, Jul 06 2016 CROSSREFS Cf. A268545-A268555. Sequence in context: A212731 A316146 A113265 * A123034 A166619 A323208 Adjacent sequences:  A124432 A124433 A124434 * A124436 A124437 A124438 KEYWORD nonn AUTHOR Lee A. Newberg, Dec 15 2006 EXTENSIONS More terms from Mark van Hoeij, Dec 21 2013 STATUS approved

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)