A127452 Triangle, read by rows of n*(n+1)/2 + 1 terms, generated by the recurrence: start with a single '1' in row 0; row n+1 is generated from row n by first inserting zeros at positions {(m+1)*(m+2)/2 - 1, m>=0} in row n and then taking the partial sums in reverse order.

1, 1, 1, 2, 2, 1, 1, 6, 6, 4, 4, 2, 1, 1, 24, 24, 18, 18, 12, 8, 8, 4, 2, 1, 1, 120, 120, 96, 96, 72, 54, 54, 36, 24, 16, 16, 8, 4, 2, 1, 1, 720, 720, 600, 600, 480, 384, 384, 288, 216, 162, 162, 108, 72, 48, 32, 32, 16, 8, 4, 2, 1, 1

Offset

0,4

Comments

The first column equals the factorials. Triangle A127420 is generated by a similar recurrence.

Links

Paul D. Hanna, Rows n = 0..30, flattened.

Formula

Sum_{k=0..n*(n+1)/2} k*T(n,k) = A018927(n+1) = Sum_{k=0..n} k*k!*{(k+1)^(n-k+1)-k^(n-k+1)}.

T(n,k) = (n-t)! * (n-t)^(k - t*(t+1)/2) * (n-t+1)^(t-k + t*(t+1)/2) where t=floor((sqrt(8*k+1)-1)/2). Also, Sum_{j=k*(k+1)/2..(k+1)*(k+2)/2-1} T(n,j) = A047969(n-k,k) = (n-k)!*((n-k+1)^(k+1)-(n-k)^(k+1)).

Example

The triangle begins:
1;
1, 1;
2, 2, 1, 1;
6, 6, 4, 4, 2, 1, 1;
24, 24, 18, 18, 12, 8, 8, 4, 2, 1, 1;
120, 120, 96, 96, 72, 54, 54, 36, 24, 16, 16, 8, 4, 2, 1, 1;
720, 720, 600, 600, 480, 384, 384, 288, 216, 162, 162, 108, 72, 48, 32, 32, 16, 8, 4, 2, 1, 1;
...
The recurrence is illustrated by the following examples.
Start with a single '1' in row 0.
To get row 1, insert 0 in row 0 at position 0,
and take partial sums in reverse order:
0,_1;
1,_1;
To get row 2, insert 0 in row 1 at positions [0,2],
and take partial sums in reverse order:
0,_1,_0,_1;
2,_2,_1,_1;
To get row 3, insert 0 in row 2 at positions [0,2,5],
and take partial sums in reverse order:
0,_2,_0,_2,_1,_0,_1;
6,_6,_4,_4,_2,_1,_1;
To get row 4, insert 0 in row 3 at positions [0,2,5,9],
and take partial sums in reverse order:
_0,__6,__0,__6,__4,_0,_4,_2,_1,_0,_1;
24,_24,_18,_18,_12,_8,_8,_4,_2,_1,_1;
etc.
Continuing in this way generates the factorials in the first column.

Prog

(PARI) T(n, k)=if(n<0 || k<0, 0, if(n==0 && k==0, 1, if(k==0, n!, if(issquare(8*k+1), T(n, k-1), T(n, k-1)-T(n-1, k-(sqrtint(8*k+1)+1)\2)))))
(PARI) T(n, k)=local(t=(sqrtint(8*k+1)-1)\2); (n-t)!*(n-t)^(k-t*(t+1)/2)*(n-t+1)^(t-k+t*(t+1)/2)

Crossrefs

Cf. A018927, A127420, A047969, A182961 (variant).

Sequence in context: A221916 A124773 A129177 * A263755 A135879 A176224

Adjacent sequences: A127449 A127450 A127451 * A127453 A127454 A127455

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 15 2007

Status

approved