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Number of lattice paths starting at {n}^6 and ending when any component equals 0, using steps that decrement one or more components by one.
2

%I #10 Apr 25 2020 14:48:00

%S 1,63,580693,39991899123,4727954015135121,716137204351882049583,

%T 125076804896889941384267749,23963247580553153291287896467139,

%U 4899254403362236213345570748744318209,1051032705565051909388116876876306460192223

%N Number of lattice paths starting at {n}^6 and ending when any component equals 0, using steps that decrement one or more components by one.

%H Alois P. Heinz, <a href="/A263164/b263164.txt">Table of n, a(n) for n = 0..50</a>

%p g():= seq(convert(n, base, 2)[1..6], n=65..127):

%p b:= proc(l) option remember;

%p `if`(l[1]=0, 1, add(b(sort(l-h)), h=g()))

%p end:

%p a:= n-> b([n$6]):

%p seq(a(n), n=0..10);

%t g[] = Table[Reverse[IntegerDigits[n, 2]][[;; 6]], {n, 2^6 + 1, 2^7 - 1}];

%t b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[]}]];

%t a[n_] := b[Table[n, {6}]];

%t a /@ Range[0, 10] (* _Jean-François Alcover_, Apr 25 2020, after _Alois P. Heinz_ *)

%Y Column k=6 of A263159.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 11 2015