0,3

For n>0, a(n) equals the element in the upper left corner of the matrix equal to the product of nXn 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 (pauldhanna(AT)juno.com), Feb 08 2007

See Maple code for precise description.

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(n)=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(n-1)=a(4)=51 will be in the upper right corner.

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:

(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 (pauldhanna(AT)juno.com), Feb 08 2007

See the triangles in A059431 and A059432.

Sequence in context: A086922 A007128 A013085 this_sequence A018987 A030518 A123188

Adjacent sequences: A059426 A059427 A059428 this_sequence A059430 A059431 A059432

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2001