A229465 Number of lattice paths from {2}^n to {0}^n using steps that decrement one component or all components by the same positive integer.

1, 2, 22, 248, 6506, 292442, 19450082, 1781791202, 214899390722, 33007840951682, 6290830043769602, 1456812593474515202, 402910665233497344002, 131173228963457333452802, 49656810289226589524275202, 21628258853895326260083456002, 10739534026001485870629015552002

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n+1/2) / exp(2*n-1). - Vaclav Kotesovec, Jul 16 2014

Maple

a:= proc(n) option remember; `if`(n<5, [1, 2, 22, 248, 6506][n+1],
     ((64481193996*n^5 -656050382562*n^4 +1835465682464*n^3
      -3691825299357*n^2 +10428520019257*n -9978603085078)*a(n-1)
      -(64481193996*n^6 -251022627918*n^5 -4253631972584*n^4
      +29686486719123*n^3 -71916661134305*n^2 +77149141951487*n
      -30090569866279)*a(n-2) +(n-2)*(437268351642*n^5
      -5777340617365*n^4 +26203609431616*n^3 -50411340883791*n^2
      +38226810988733*n -9795152028455)*a(n-3) -(n-2)*(n-3)*
      (170273280324*n^4 -2136687453608*n^3 +8692120865702*n^2
      -11643795721897*n +4287224601259)*a(n-4) -(n-6)*(n-2)*(n-3)*
      (n-4)*(202513877322*n^2-310611483677*n+98391999767)*a(n-5))/
      (32240596998*n^3-328025191281*n^2+768115007074*n-189524735891))
    end:
seq(a(n), n=0..20);

Mathematica

b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}], 0] + Sum[Sum[b[Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]];
a[k_] := b[Array[2&, k]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz in A229345 *)

Crossrefs

Row n=2 of A229345.

Sequence in context: A334603 A342232 A082777 * A072076 A226706 A036841

Adjacent sequences: A229462 A229463 A229464 * A229466 A229467 A229468

Keyword

nonn

Author

Alois P. Heinz, Sep 24 2013

Status

approved