login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A090214 Generalized Stirling2 array S_{4,4}(n,k). 9
1, 24, 96, 72, 16, 1, 576, 13824, 50688, 59904, 30024, 7200, 856, 48, 1, 13824, 1714176, 21606912, 76317696, 110160576, 78451200, 30645504, 6976512, 953424, 78400, 3760, 96, 1, 331776, 207028224, 8190885888, 74684104704, 253100173824 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The row length sequence for this array is [1,5,9,13,17,...] = A016813(n-1), n >= 1.
The g.f. for the k-th column, (with leading zeros and k >= 4) is G(k,x) = x^ceiling(k/4)*P(k,x)/Product_{p = 4..k} (1 - fallfac(p,4)*x), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := Sum_{m = 0..kmax(k)} A090221(k,m)*x^m, k >= 4, with kmax(k) := A057353(k-4)= floor(3*(k-4)/4). For the recurrence of the G(k,x) see A090221.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_4 (the disjoint union of n copies of the complete graph K_4). - Peter Bala, Aug 15 2013
LINKS
Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 71, flattened)
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Codara, O. M. D’Antona, and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv:1308.1700v1 [cs.DM], 2013.
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - N. J. A. Sloane, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Wolfdieter Lang, First 4 rows.
FORMULA
a(n, k) = (-1)^k/k! * Sum_{p = 4..k} (-1)^p * binomial(k, p) * fallfac(p, 4)^n, with fallfac(p, 4) := A008279(p, 4) = p*(p - 1)*(p - 2)*(p - 3); 4 <= k <= 4*n, n >= 1, else 0. From eq.(19) with r = 4 of the Blasiak et al. reference.
E^n = Sum_{k = 4..4*n} a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator (x^4)*d^4/dx^4.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k >= 0} (k*(k - 1)*(k - 2)*(k - 3))^n*x^k/k!. - Peter Bala, Aug 15 2013
EXAMPLE
Table begins
n\k| 4 5 6 7 8 9 10 11 12
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
1 | 1
2 | 24 96 72 16 1
3 | 576 13824 50688 59904 30024 7200 856 48 1
...
MAPLE
T:= (n, k) -> (-1)^k/k!*add((-1)^p*binomial(k, p)*(p*(p-1)*(p-2)*(p-3))^n, p=4..k):
seq(seq(T(n, k), k=4..4*n), n=1..10); # Robert Israel, Jan 28 2016
MATHEMATICA
a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*FactorialPower[p, 4]^n, {p, 4, k}]; Table[a[n, k], {n, 1, 5}, {k, 4, 4*n}] // Flatten (* Jean-François Alcover, Sep 05 2012, updated Jan 28 2016 *)
CROSSREFS
Cf. A090215, A071379 (row sums), A090213 (alternating row sums).
S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{2, 2} = A078739, S_{3, 2} = A078740, S_{3, 3} = A078741.
Sequence in context: A365191 A353326 A055671 * A283446 A208984 A103251
KEYWORD
nonn,easy,tabf,changed
AUTHOR
Wolfdieter Lang, Dec 01 2003
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 07:20 EDT 2024. Contains 371235 sequences. (Running on oeis4.)