A207645 Triangle where T(n,k) = Product_{j=1..k} floor(n/j - 1), as read by rows n>=0, columns k=0..[n/2].

1, 1, 1, 1, 1, 2, 1, 3, 3, 1, 4, 4, 1, 5, 10, 10, 1, 6, 12, 12, 1, 7, 21, 21, 21, 1, 8, 24, 48, 48, 1, 9, 36, 72, 72, 72, 1, 10, 40, 80, 80, 80, 1, 11, 55, 165, 330, 330, 330, 1, 12, 60, 180, 360, 360, 360, 1, 13, 78, 234, 468, 468, 468, 468, 1, 14, 84, 336, 672, 1344, 1344, 1344

Offset: 0

Author

Paul D. Hanna, Feb 20 2012

Keywords

• nonn
• nice
• tabf

Comments

Compare the definition to that of Pascal's triangle:

binomial(n,k) = Product_{j=1..k} ((n+1)/j - 1).

Examples

			Triangle begins with row n=0 as:
1;
1;
1,  1;
1,  2;
1,  3,   3;
1,  4,   4;
1,  5,  10,  10;
1,  6,  12,  12;
1,  7,  21,  21,   21;
1,  8,  24,  48,   48;
1,  9,  36,  72,   72,   72;
1, 10,  40,  80,   80,   80;
1, 11,  55, 165,  330,  330,  330;
1, 12,  60, 180,  360,  360,  360;
1, 13,  78, 234,  468,  468,  468,  468;
1, 14,  84, 336,  672, 1344, 1344, 1344;
1, 15, 105, 420, 1260, 2520, 2520, 2520, 2520;
1, 16, 112, 448, 1344, 2688, 2688, 2688, 2688; ...
		

Links

• Paul D. Hanna, Rows n = 0..70, flattened

Crossrefs

Cf. A207643, A207644, A207646, A207647.

Programs

Mathematica

t[n_, k_] := Product[Floor[n/j - 1], {j, 1, k}]; Flatten[Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}]] (* Jean-François Alcover, Jun 12 2012 *)

PARI

{T(n,k)=if(k==0,1,prod(j=1,k,floor(n/j-1)))}
for(n=0,12,for(k=0,n\2,print1(T(n,k),", "));print(""))

Formula

Row sums equal A207643.

Antidiagonal sums form A207644.

Right border of even-indexed rows equals A207646.

Right border of odd-indexed rows equals A207647.