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.

1, 15, 2101, 717795, 328504401, 172924236255, 98788351385893, 59547100211425779, 37279994808479614465, 24006888102075722880975, 15800133137207909144690421, 10580854797781352259168325347, 7186571606168294602440625922385, 4938826696886704892539811529645855

Offset: 0

Author

Alois P. Heinz, Oct 11 2015

Keywords

• nonn

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..263 (terms 0..170 from Alois P. Heinz)

Crossrefs

Column k=4 of A263159.

Cf. A263064.

Programs

Maple

g():= seq(convert(n, base, 2)[1..4], n=17..31):
b:= proc(l) option remember;
      `if`(l[1]=0, 1, add(b(sort(l-h)), h=g()))
    end:
a:= n-> b([n$4]):
seq(a(n), n=0..16);

Mathematica

g[] = Table[Reverse[IntegerDigits[n, 2]][[;; 4]], {n, 2^4 + 1, 2^5 - 1}];
b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[]}]];
a[n_] := b[Table[n, {4}]];
a /@ Range[0, 16] (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz *)

Formula

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