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Number of lattice paths starting at {n}^4 and ending when any component equals 0, using steps that decrement one or more components by one.
2

%I #19 Apr 25 2020 14:47:48

%S 1,15,2101,717795,328504401,172924236255,98788351385893,

%T 59547100211425779,37279994808479614465,24006888102075722880975,

%U 15800133137207909144690421,10580854797781352259168325347,7186571606168294602440625922385,4938826696886704892539811529645855

%N Number of lattice paths starting at {n}^4 and ending when any component equals 0, using steps that decrement one or more components by one.

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A263162/b263162.txt">Table of n, a(n) for n = 0..263</a> (terms 0..170 from Alois P. Heinz)

%F a(n) ~ c * d^n / (Pi^(3/2) * n^(3/2)), where d = 195 + 138*sqrt(2) + 4*sqrt(4756 + 3363*sqrt(2)) = 780.279406806795145659... and c = sqrt(112232 - 176706*sqrt(2) + sqrt(-24823369828 + 32297875299*sqrt(2)))/2744 = 0.02991158822483794318293134... . - _Vaclav Kotesovec_, Nov 28 2016

%p g():= seq(convert(n, base, 2)[1..4], n=17..31):

%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$4]):

%p seq(a(n), n=0..16);

%t g[] = Table[Reverse[IntegerDigits[n, 2]][[;; 4]], {n, 2^4 + 1, 2^5 - 1}];

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

%t a[n_] := b[Table[n, {4}]];

%t a /@ Range[0, 16] (* _Jean-François Alcover_, Apr 25 2020, after _Alois P. Heinz_ *)

%Y Column k=4 of A263159.

%Y Cf. A263064.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 11 2015