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A263162 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.

%I A263162 #19 Apr 25 2020 14:47:48
%S A263162 1,15,2101,717795,328504401,172924236255,98788351385893,
%T A263162 59547100211425779,37279994808479614465,24006888102075722880975,
%U A263162 15800133137207909144690421,10580854797781352259168325347,7186571606168294602440625922385,4938826696886704892539811529645855
%N A263162 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 A263162 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 A263162 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 A263162 g():= seq(convert(n, base, 2)[1..4], n=17..31):
%p A263162 b:= proc(l) option remember;
%p A263162       `if`(l[1]=0, 1, add(b(sort(l-h)), h=g()))
%p A263162     end:
%p A263162 a:= n-> b([n$4]):
%p A263162 seq(a(n), n=0..16);
%t A263162 g[] = Table[Reverse[IntegerDigits[n, 2]][[;; 4]], {n, 2^4 + 1, 2^5 - 1}];
%t A263162 b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[]}]];
%t A263162 a[n_] := b[Table[n, {4}]];
%t A263162 a /@ Range[0, 16] (* _Jean-François Alcover_, Apr 25 2020, after _Alois P. Heinz_ *)
%Y A263162 Column k=4 of A263159.
%Y A263162 Cf. A263064.
%K A263162 nonn
%O A263162 0,2
%A A263162 _Alois P. Heinz_, Oct 11 2015