A090438 - OEIS

%I #24 Jan 13 2025 06:36:51

%S 1,12,8,1,360,480,180,24,1,20160,40320,25200,6720,840,48,1,1814400,

%T 4838400,4233600,1693440,352800,40320,2520,80,1,239500800,798336000,

%U 898128000,479001600,139708800,23950080,2494800,158400,5940,120,1

%N Generalized Stirling2 array (4,2).

%C The row length sequences for this array is [1,3,5,7,9,11,...] = A005408(n-1), n>=1.

%C 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).

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.

%H Wolfdieter Lang, <a href="/A090438/a090438.txt">First 6 rows</a>.

%H M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.

%F 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).

%F 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.

%F a(n, k) = ((2*n)!/k!)*binomial(2*(n-1), k-2), n>=1, 2<=k<=2*n.

%F 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)).

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

%F Coefficient triangle of Laguerre polynomials (2*n)!*L(2*n,2,-x). - _Peter Luschny_, Apr 08 2015

%p with(PolynomialTools):

%p p := n -> (2*n+2)!*hypergeom([-2*n],[3], -x)/2:

%p seq(CoefficientList(simplify(p(n)),x), n=0..5); # _Peter Luschny_, Apr 08 2015

%t 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 *)

%Y Cf. A078740 (3, 2)-Stirling2.

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

%Y Cf. A062139.

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Dec 23 2003