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A091534 Generalized Stirling2 array (5,2). 11
1, 20, 10, 1, 1120, 1040, 290, 30, 1, 123200, 161920, 71320, 14040, 1340, 60, 1, 22422400, 37452800, 22097600, 6263040, 958720, 82800, 4000, 100, 1, 6098892800, 12222918400, 8928102400, 3257116800, 675281600, 84782880, 6625920, 322000 (list; graph; refs; listen; history; text; 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.
LINKS
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
W. Lang, First 6 rows.
FORMULA
a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+3*(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=5, s=2.
Recursion: a(n, k)=sum(binomial(2, p)*fallfac(3*(n-1)+k-p, 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=5, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
MATHEMATICA
a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 3*(j - 1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2 n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)
CROSSREFS
Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2.
Cf. A072019 (row sums), A091537 (alternating row sums).
Sequence in context: A078080 A216289 A136010 * A033966 A033340 A040383
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Jan 23 2004
STATUS
approved

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Last modified February 22 14:47 EST 2024. Contains 370256 sequences. (Running on oeis4.)