A059429 Cumulative boustrophedon transform of 1, 0, 0, 0, ...

1, 1, 2, 8, 51, 478, 6178, 105330, 2290069, 61839897, 2030449500, 79661186168, 3680458880352, 197781841355220, 12231649482909444, 862560715175755168, 68799732139319891208, 6162698115430291654438, 615995773861169229993018

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

Links

Table of n, a(n) for n=0..18.

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

See the triangles in A059431 and A059432.

Sequence in context: A352147 A351772 A277506 * A249747 A191480 A013555

Adjacent sequences: A059426 A059427 A059428 * A059430 A059431 A059432

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jan 31 2001

Status

approved