|
| |
|
|
A090438
|
|
Generalized Stirling2 array (4,2).
|
|
14
| |
|
|
1, 12, 8, 1, 360, 480, 180, 24, 1, 20160, 40320, 25200, 6720, 840, 48, 1, 1814400, 4838400, 4233600, 1693440, 352800, 40320, 2520, 80, 1, 239500800, 798336000, 898128000, 479001600, 139708800, 23950080, 2494800, 158400, 5940, 120, 1
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| The row length sequences for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.
The scaled array a(n,k)/((2*n)!/k!) = A034870(n-1,k-2), n>=1, 2<=k<=2*n (Pascal triangle, even numbered rows only).
|
|
|
REFERENCES
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
|
|
|
LINKS
| W. Lang, First 6 rows.
|
|
|
FORMULA
| Recursion: a(n, k)=sum(binomial(2, p)*fallfac(2*(n-1)-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, 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+2*(j-1), 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=4, s=2.
a(n, k)=((2*n)!/k!)*binomial(2*(n-1), k-2), n>=1, 2<=k<=2*n.
E.g.f. column k>=2 (with leading zeros): (((-1)^k)/k!)*(sum(((-1)^p)*binomial(k, p)*hypergeom([(p-1)/2, p/2], [], 4*x), p=2..k)-(k-1)).
|
|
|
CROSSREFS
| Cf. A078740 (3, 2)-Stirling2.
Cf. A072678 (row sums), A090439 (alternating row sums).
Sequence in context: A206423 A038333 A185259 * A128108 A071279 A118656
Adjacent sequences: A090435 A090436 A090437 * A090439 A090440 A090441
|
|
|
KEYWORD
| nonn,easy,tabf
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 23 2003
|
| |
|
|
