A263162 - OEIS

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

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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 *)