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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.
6
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
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 15 2007
STATUS
approved