|
|
A078740
|
|
Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n).
|
|
19
|
|
|
1, 6, 6, 1, 72, 168, 96, 18, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 43200, 259200, 424800, 285120, 92520, 15600, 1380, 60, 1, 1814400, 15120000, 34776000, 33566400, 16304400, 4379760, 682200, 62400, 3270, 90, 1, 101606400, 1117670400
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The sequence of row lengths of this array is [1,3,5,7,...] = A005408(n-1), n>=1.
For the scaled array s2_{3,2}(n,k) := a(n,k)*k!/((n+1)!*n!) see A090452.
|
|
LINKS
|
|
|
FORMULA
|
Recursion: a(n, k) = Sum(binomial(2, p)*fallfac(n-1-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 1)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
a(n, k) = (((-1)^k)/k!)*Sum(((-1)^p)*binomial(k, p)*product(fallfac(p+(j-1)*(3-2), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=3, s=2.
|
|
EXAMPLE
|
1;
6, 6, 1;
72, 168, 96, 18, 1;
...
|
|
MATHEMATICA
|
a[n_, k_] := (-1)^k*n!*(n+1)!*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!); Table[a[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|