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

0,2

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

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

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 *)

Crossrefs

Column k=4 of A262809.

Sequence in context: A251242 A259464 A116527 * A085404 A110100 A003745

Adjacent sequences: A263061 A263062 A263063 * A263065 A263066 A263067

Keyword

nonn

Author

Alois P. Heinz, Oct 08 2015

Status

approved