A263064 Number of lattice paths from (n,n,n,n) to (0,0,0,0) using steps that decrement one or more components by one.

1, 75, 23917, 10681263, 5552351121, 3147728203035, 1887593866439485, 1177359342144641535, 756051015055329306625, 496505991344667030490635, 331910222316215755702672557, 225110028217225196478861017775, 154515942591851050758389232988689

Offset: 0

Author

Alois P. Heinz, Oct 08 2015

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350
• J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Crossrefs

Column k=4 of A262809.

Programs

Mathematica

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 *)

Formula

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

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