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

0,2

Links

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

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 *)

Crossrefs

Column k=4 of A263159.

Cf. A263064.

Sequence in context: A199098 A208784 A126681 * A232196 A222881 A229844

Adjacent sequences: A263159 A263160 A263161 * A263163 A263164 A263165

Keyword

nonn

Author

Alois P. Heinz, Oct 11 2015

Status

approved