A229482 Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component or all components by the same positive integer.

1, 7, 248, 11380, 577124, 30970588, 1724240804, 98508192580, 5736813639188, 339068764626556, 20277072462706100, 1224258843324348388, 74504869395134442884, 4564559749008113090620, 281250580532881468554692, 17415330397418786646707236

Offset: 0

Author

Alois P. Heinz, Sep 24 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Vaclav Kotesovec, Recurrence (of order 6)

Crossrefs

Column k=3 of A229345.

Programs

Maple

b:= proc(l) local m; m:= nops(l); if m=0 or l[m]=0 then 1
      elif m>1 then b(l):= add(add(b(sort(subsop(i=l[i]-j, l))),
      j=1..l[i]), i=1..m)+add(b(map(x->x-j, l)), j=1..l[1]) else 0 fi
    end:
a:= n-> b([n$3]):
seq(a(n), n=0..20);

Mathematica

b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}], 0] + Sum[Sum[b[Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]];
a[n_] := b[{n, n, n}];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

Formula

a(n) ~ c*d^n/n, where d = (3*(375+sqrt(17))^(2/3)+156+23*(375+sqrt(17))^(1/3))/(375+sqrt(17))^(1/3) = 66.266905910039023... is the root of the equation -125 + 183*d - 69*d^2 + d^3 = 0 and c = sqrt(-269/225 + 2*sqrt(14561) * cosh(arccosh(60154403/(116488*sqrt(14561)))/3)/225)/Pi = 0.1272434612906147722352211214089... - Vaclav Kotesovec, Sep 25 2013, updated Mar 17 2024