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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. 2
1, 2, 22, 248, 6506, 292442, 19450082, 1781791202, 214899390722, 33007840951682, 6290830043769602, 1456812593474515202, 402910665233497344002, 131173228963457333452802, 49656810289226589524275202, 21628258853895326260083456002, 10739534026001485870629015552002 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified September 20 00:20 EDT 2021. Contains 347577 sequences. (Running on oeis4.)