A090438 Generalized Stirling2 array (4,2).

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

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).

Links

Table of n, a(n) for n=1..36.

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

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)).

Coefficient triangle of the polynomials (2*n+2)!*hypergeom([-2*n],[3],-x)/2. - Peter Luschny, Apr 08 2015

Coefficient triangle of Laguerre polynomials (2*n)!*L(2*n,2,-x)). - Peter Luschny, Apr 08 2015

Maple

with(PolynomialTools):
p := n -> (2*n+2)!*hypergeom([-2*n], [3], -x)/2:
seq(CoefficientList(simplify(p(n)), x), n=0..5); # Peter Luschny, Apr 08 2015

Mathematica

a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 2*(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.

Cf. A072678 (row sums), A090439 (alternating row sums).

Cf. A062139.

Sequence in context: A319406 A038333 A185259 * A346223 A128108 A071279

Adjacent sequences: A090435 A090436 A090437 * A090439 A090440 A090441

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Dec 23 2003

Status

approved