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

Author

Alois P. Heinz, Sep 24 2013

Keywords

• nonn

Links

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

Crossrefs

Row n=2 of A229345.

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [1, 2, 22, 248, 6506][n+1],
     ((64481193996*n^5 -656050382562*n^4 [phone]*n^3
      -3691825299357*n^2 [phone]*n -9978603085078)*a(n-1)
      -(64481193996*n^6 -251022627918*n^5 -4253631972584*n^4
      [phone]*n^3 -71916661134305*n^2 [phone]*n
      -30090569866279)*a(n-2) +(n-2)*(437268351642*n^5
      -5777340617365*n^4 [phone]*n^3 -50411340883791*n^2
      [phone]*n -9795152028455)*a(n-3) -(n-2)*(n-3)*
      (170273280324*n^4 -2136687453608*n^3 [phone]*n^2
      -11643795721897*n [phone])*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 *)

Formula

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