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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. 2
1, 7, 248, 11380, 577124, 30970588, 1724240804, 98508192580, 5736813639188, 339068764626556, 20277072462706100, 1224258843324348388, 74504869395134442884, 4564559749008113090620, 281250580532881468554692, 17415330397418786646707236 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vaclav Kotesovec, Recurrence (of order 6)
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
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 *)
CROSSREFS
Column k=3 of A229345.
Sequence in context: A203158 A227414 A203698 * A193371 A347843 A338633
KEYWORD
nonn,changed
AUTHOR
Alois P. Heinz, Sep 24 2013
STATUS
approved

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Last modified March 28 14:13 EDT 2024. Contains 371254 sequences. (Running on oeis4.)