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A059429 Cumulative boustrophedon transform of 1, 0, 0, 0, ... 2
1, 1, 2, 8, 51, 478, 6178, 105330, 2290069, 61839897, 2030449500, 79661186168, 3680458880352, 197781841355220, 12231649482909444, 862560715175755168, 68799732139319891208, 6162698115430291654438, 615995773861169229993018 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A007128 A013085 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

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Last modified September 18 07:39 EDT 2020. Contains 337166 sequences. (Running on oeis4.)