OFFSET
0,3
COMMENTS
For n>0, a(n) equals the element in the upper left corner of the matrix equal to the product of n X n matrices given by: Product_{k=1..n} M_k where M_k(r,c)=k+2-r-c when r+c<=k+1 and zeros elsewhere (see example). - Paul D. Hanna, Feb 08 2007
FORMULA
See Maple code for precise description.
EXAMPLE
For n=5, the matrix product:
[1 0 0 0 0] [2 1 0 0 0] [3 2 1 0 0] [4 3 2 1 0] [5 4 3 2 1]
[0 0 0 0 0] [1 0 0 0 0] [2 1 0 0 0] [3 2 1 0 0] [4 3 2 1 0]
[0 0 0 0 0] [0 0 0 0 0] [1 0 0 0 0] [2 1 0 0 0] [3 2 1 0 0]
[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [1 0 0 0 0] [2 1 0 0 0]
[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [1 0 0 0 0]
equals the matrix below with a(5)=478 in the upper left corner:
[478 362 246 138 51]
[__0 __0 __0 __0 _0]
[__0 __0 __0 __0 _0]
[__0 __0 __0 __0 _0]
[__0 __0 __0 __0 _0]
also a(4)=51 will be in the upper right corner.
MAPLE
CBOUS2 := proc(a) option remember; local c, i, j, n, r: if whattype(a) <> list then RETURN([]); fi: n := min( nops(a), 60); for i from 0 to n-1 do c[i, 0] := a[i+1]; od; for i to n-1 do for j to i do c[i, j] := c[i, j-1] + add(c[i-1, i-r], r=1..j); od; od; RETURN([seq(c[i, i], i=0..n-1)]); end:
MATHEMATICA
m[n_, k_] := Table[If[r+c <= k+1, k+2-r-c, 0], {r, 1, n}, {c, 1, n}]; a[0] = 1; a[n_] := (Dot @@ Table[m[n, k], {k, 1, n}])[[1, 1]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 18 2012, after Paul D. Hanna *)
PROG
(PARI) {a(n)=if(n==0, 1, prod(k=1, n, matrix(n, n, r, c, if(r+c<=k+1, k+2-r-c)))[1, 1])} \\ Paul D. Hanna, Feb 08 2007
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jan 31 2001
STATUS
approved