A263067 Number of lattice paths from {n}^7 to {0}^7 using steps that decrement one or more components by one.

1, 47293, 58514835289, 143743469278461361, 480086443888959812703121, 1909946024633189859690880523893, 8508048612432263410111274212273801489, 41020870889694863957061607086939138327565057, 209691630817770382144439647416526247292909726379393

Offset: 0

Author

Alois P. Heinz, Oct 08 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Vaclav Kotesovec, Recurrence (of order 7)

Crossrefs

Column k=7 of A262809.

Programs

Mathematica

With[{k = 7}, 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

a(n) ~ sqrt(c) * d^n / (Pi*n)^3, where d = 7553550.61983382187210690975164995019966376572879... is the root of the equation -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7 = 0 and c = 0.1137319057755565367034882185733003109119819... is the root of the equation -1 - 12544*c - 61816832*c^2 - 151057858560*c^3 - 189486977777664*c^4 - 113186888059191296*c^5 - 25353862925258850304*c^6 + 231806746745223774208*c^7 = 0. - Vaclav Kotesovec, Mar 23 2016