A109876 Triangle read by rows: a(n, n) = n! and for 1 <= k < n, a(n, k) = Sum_{i=0..n-1} Product_{j=i+1..i+k} f(j, n), where for x <= y, f(x, y) = x and for x > y, f(x, y) = x-y.

1, 3, 2, 6, 11, 6, 10, 24, 50, 24, 15, 45, 120, 274, 120, 21, 76, 252, 720, 1764, 720, 28, 119, 476, 1680, 5040, 13068, 5040, 36, 176, 828, 3520, 12960, 40320, 109584, 40320, 45, 249, 1350, 6750, 29880, 113400, 362880, 1026576, 362880, 55, 340, 2090, 12048

Offset

1,2

Comments

The first four columns (excluding the initial term of each) are A000217 (triangular numbers), A006527, A062026 and A062027. The first and third diagonals are both A000142 (factorials). The second diagonal is A000254.

Without the exception for k = n, a(n, n) would be n*n! (A001563(n)). For example, a(3, 3) would be 1*2*3 + 2*3*1 + 3*1*2 instead of 1*2*3. The author's original description did not mention the exception. I guess it didn't make sense to him to add n identical terms. - David Wasserman, Oct 01 2008

Links

Table of n, a(n) for n=1..49.

Example

a(5, 3) = 1*2*3 + 2*3*4 + 3*4*5 + 4*5*1 + 5*1*2 = 120.

Prog

(PARI) f(x, y) = if (x > y, x - y, x);
a(n, k) = if (n == k, n!, sum (i = 0, n - 1, prod (j = i + 1, i + k, f(j, n)))); \\ David Wasserman, Oct 01 2008

Crossrefs

Cf. A109877.

Sequence in context: A340612 A362373 A364929 * A108284 A340733 A095011

Adjacent sequences: A109873 A109874 A109875 * A109877 A109878 A109879

Keyword

nonn,easy,tabl

Author

Amarnath Murthy, Jul 10 2005

Extensions

Edited and extended by David Wasserman, Oct 01 2008

Status

approved