A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 6, 2, 0, 1, 1, 24, 54, 2, 0, 1, 1, 120, 1944, 384, 2, 0, 1, 1, 720, 99000, 132000, 2550, 2, 0, 1, 1, 5040, 6966000, 79716000, 8059800, 16506, 2, 0, 1, 1, 40320, 655678800, 78928416000, 57010275000, 471369024, 105840, 2, 0, 1
Offset: 0
A225050 Number of shortest paths from one vertex of a cube (side = n units) to farthest vertex, along the grid on 3 surfaces meeting at another vertex.
1, 4, 24, 148, 920, 5754, 36204, 229128, 1458072, 9325030, 59905274, 386368008, 2500651244, 16234450288, 105680341560, 689579733648, 4509096746904, 29539804567230, 193843622021910, 1273932593379600, 8383551165082170, 55238332546851630, 364364231864557080
Offset: 0
Crossrefs
Cf. A060774.
Programs
-
Maple
A225050:=n->2*binomial(3*n,n)-binomial(2*n,n); seq(A225050(n), n=0..30); # Wesley Ivan Hurt, May 10 2014
-
Mathematica
Table[2*Binomial[3 n, n] - Binomial[2 n, n], {n, 30}] -
PARI
a(n)=2*binomial(3*n,n)-binomial(2*n,n) \\ Charles R Greathouse IV, Apr 26 2013
Formula
a(n) = 2*binomial(3n,n) - binomial(2n,n).
Comments
Examples
A(n,0) = 1: [()]. A(0,k) = 1: [{0}^k]. A(1,1) = 1: [(1), (0)]. A(2,1) = 0, there is no path from (2) to (0) without interior points. A(1,2) = 2: [(1,1), (0,1), (0,0)], [(1,1), (1,0), (0,0)]. A(1,3) = 6: [(1,1,1), (0,1,1), (0,0,1), (0,0,0)], [(1,1,1), (0,1,1), (0,1,0), (0,0,0)], [(1,1,1), (1,0,1), (0,0,1), (0,0,0)], [(1,1,1), (1,0,1), (1,0,0), (0,0,0)], [(1,1,1), (1,1,0), (0,1,0), (0,0,0)], [(1,1,1), (1,1,0), (1,0,0), (0,0,0)]. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 1, 1, 2, 6, 24, 120, ... 1, 0, 2, 54, 1944, 99000, ... 1, 0, 2, 384, 132000, 79716000, ... 1, 0, 2, 2550, 8059800, 57010275000, ... 1, 0, 2, 16506, 471369024, 38606650125120, ...Links
Crossrefs
Programs
Maple
b:= proc(n, l) option remember; local m; m:= nops(l); `if`(m=0 or l[m]=0, 1, `if`(l[1]>0 and l[m] b(n, [n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10); Mathematica
b[n_, l_] := b[n, l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[l[[1]] > 0 && l[[m]] < n, 0, Sum[If[l[[i]] == 0, 0, b[n, Sort[ReplacePart[l, i -> l[[i]] - 1]]]], {i, 1, m}]]] ]; a[n_, k_] := b[n, Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)