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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 Table of n, a(n) for n=1..33. 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. M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665. 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 Adjacent sequences: A091531 A091532 A091533 * A091535 A091536 A091537 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.)