A263064 - OEIS

%I #13 Sep 09 2016 12:23:53

%S 1,75,23917,10681263,5552351121,3147728203035,1887593866439485,

%T 1177359342144641535,756051015055329306625,496505991344667030490635,

%U 331910222316215755702672557,225110028217225196478861017775,154515942591851050758389232988689

%N Number of lattice paths from (n,n,n,n) to (0,0,0,0) using steps that decrement one or more components by one.

%C Also, the number of alignments for 4 sequences of length n each (Slowinski 1998).

%H Alois P. Heinz, <a href="/A263064/b263064.txt">Table of n, a(n) for n = 0..350</a>

%H J. B. Slowinski, <a href="http://www.neurociencias.org.ve/cont-cursos-laboratorio-de-neurociencias-luz/Slowinski1998%20phylogenetics.pdf">The Number of Multiple Alignments</a>, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:<a href="http://dx.doi.org/10.1006/mpev.1998.0522">10.1006/mpev.1998.0522</a>

%F Recurrence: (n-1)*n^3*(864*n^4 - 6480*n^3 + 17763*n^2 - 21015*n + 9059)*a(n) = 15*(n-1)*(44928*n^7 - 404352*n^6 + 1459788*n^5 - 2712556*n^4 + 2772389*n^3 - 1538829*n^2 + 423093*n - 43506)*a(n-1) + (188352*n^8 - 2166048*n^7 + 10541118*n^6 - 28166748*n^5 + 44769259*n^4 - 42719172*n^3 + 23364582*n^2 - 6470217*n + 671094)*a(n-2) + 3*(n-2)*(3456*n^7 - 38016*n^6 + 169116*n^5 - 388336*n^4 + 486619*n^3 - 322644*n^2 + 100014*n - 10989)*a(n-3) - (n-3)^3*(n-2)*(864*n^4 - 3024*n^3 + 3507*n^2 - 1473*n + 191)*a(n-4). - _Vaclav Kotesovec_, Mar 22 2016

%F a(n) ~ sqrt(8 + 6*sqrt(2) + sqrt(140 + 99*sqrt(2))) * (195 + 138*sqrt(2) + 4*sqrt(4756 + 3363*sqrt(2)))^n / (8 * Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Mar 22 2016

%t With[{k = 4}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 15}]] (* _Vaclav Kotesovec_, Mar 22 2016 *)

%Y Column k=4 of A262809.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 08 2015