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%I #22 Mar 17 2024 07:27:38
%S 1,7,248,11380,577124,30970588,1724240804,98508192580,5736813639188,
%T 339068764626556,20277072462706100,1224258843324348388,
%U 74504869395134442884,4564559749008113090620,281250580532881468554692,17415330397418786646707236
%N Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component or all components by the same positive integer.
%H Alois P. Heinz, <a href="/A229482/b229482.txt">Table of n, a(n) for n = 0..300</a>
%H Vaclav Kotesovec, <a href="/A229482/a229482.txt">Recurrence (of order 6)</a>
%F 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
%p b:= proc(l) local m; m:= nops(l); if m=0 or l[m]=0 then 1
%p elif m>1 then b(l):= add(add(b(sort(subsop(i=l[i]-j, l))),
%p j=1..l[i]), i=1..m)+add(b(map(x->x-j, l)), j=1..l[1]) else 0 fi
%p end:
%p a:= n-> b([n$3]):
%p seq(a(n), n=0..20);
%t 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}]]];
%t a[n_] := b[{n, n, n}];
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ *)
%Y Column k=3 of A229345.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Sep 24 2013
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