A059117 Square array of lambda(k,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly k starting and/or finishing points.

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 6, 24, 1, 0, 0, 0, 0, 0, 114, 78, 1, 0, 0, 0, 0, 0, 180, 978, 240, 1, 0, 0, 0, 0, 0, 90, 4320, 6810, 726, 1, 0, 0, 0, 0, 0, 0, 8460, 63540, 43746, 2184, 1, 0, 0, 0, 0, 0, 0, 7560, 271170, 774000, 271194, 6558, 1

Offset: 0

Author

Henry Bottomley, Jan 05 2001

Keywords

• nonn
• tabl

Examples

			Rows are: 1,0,0,0,0,0,....; 0,0,1,0,0,0,....; 0,0,1,6,6,0,....; 0,0,1,24,114,180,.... etc.
		

Crossrefs

Sum of rows gives A055203. Columns include A000007, A057427, A058809, A059116. Final positive number in each row is A000680.

Programs

Mathematica

A[ n_, k_] := If[n < 1 || k < 1, Boole[n == 0 && k == 0], n! k! Coefficient[ Normal[ Series[ Sum[ Exp[-x z] (x z)^m/m! Exp[y z m (m - 1)/2], {m, 0, n}], {z, 0, n + k}]], x^n y^k z^(n + k)]]; (* Michael Somos, Jul 17 2019 *)

Formula

lambda(k, n) = (lambda(k - 2, n - 1) + 2*lambda(k - 2, n - 1) + lambda(k - 2, n - 1))*k*(k - 1)/2 starting with lambda(k, 0) = 1 if k = 0 but = 0 otherwise. lambda(k, n) = sum_{j=0..k} (-1)^(k + j) * C(k, j) * ((j - 1)*j/2)^n.