|
| |
|
|
A060475
|
|
Triangular array formed from successive differences of factorial numbers, then with factorials removed.
|
|
5
| |
|
|
1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 3, 7, 11, 9, 1, 4, 13, 32, 53, 44, 1, 5, 21, 71, 181, 309, 265, 1, 6, 31, 134, 465, 1214, 2119, 1854, 1, 7, 43, 227, 1001, 3539, 9403, 16687, 14833, 1, 8, 57, 356, 1909, 8544, 30637, 82508, 148329, 133496, 1, 9, 73, 527, 3333, 18089
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,8
|
|
|
COMMENTS
| T(n,k) is also the number of partial bijections (of an n-element set) with a fixed domain of size k and without fixed points. Equivalently, T(n,k) is the number of partial derangements with a fixed domain of size k in the symmetric inverse semigroup (monoid), I sub n. [From A. Umar, Sep 14 2008]
|
|
|
REFERENCES
| Laradji, A. and Umar, A. Combinatorial results for the symmetric inverse semigroup. Semigroup Forum 75 (2007), 221-236. [From A. Umar, Sep 14 2008]
|
|
|
LINKS
| L. Takacs, The Problem of Coincidences, Archive for History of Exact Sciences, Volume 21, No. 3, Sept. 1980. pp 229-244, paragraph 10 (Catalan).
|
|
|
FORMULA
| T(n,k) =A047920(n,k)/(n-k)! =(n-1)*T(n-1,k-1)+(k-1)*T(n-2,k-2) =(n-k+1)*T(n, k-1)-T(n-1,k-1)
T(n,k) = k! * sum(j=0..k, C(n-j,k-j)*(-1)^j/j! ). [From A. Umar, Sep 14 2008]
C(n,k)*T(n,k) = A144089(n, k) [From A. Umar, Sep 14 2008]
T(n,k) = A076732(n+1,k+1)/(k+1) [Johannes W. Meijer, Jul 27 2011]
|
|
|
MAPLE
| A060475 := proc(n, k): k! * sum(binomial(n-j, k-j)*(-1)^j/j!, j=0..k) end: seq(seq(A060475(n, k), k=0..n), n=0..7); [Johannes W. Meijer, Jul 27 2011]
|
|
|
MATHEMATICA
| t[n_, k_] := k!*Sum[Binomial[n - j, k - j]*(-1)^j/j!, {j, 0, k}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Aug 08 2011 *)
|
|
|
CROSSREFS
| Columns in one direction include A000012, A001477, A002061. Columns in other direction include A000166, A000255, A000153, A000261, A001909, A001910. Main diagonal is A002119.
Similar to A076731.
Row sums equal A003470. [Johannes W. Meijer, Jul 27 2011]
Sequence in context: A102288 A107357 A026105 * A168069 A106559 A106377
Adjacent sequences: A060472 A060473 A060474 * A060476 A060477 A060478
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Mar 16 2001
|
| |
|
|
