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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.

%I A059117 #5 Jul 17 2019 08:35:47
%S A059117 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,
%T A059117 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,
%U A059117 0,8460,63540,43746,2184,1,0,0,0,0,0,0,7560,271170,774000,271194,6558,1
%N 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.
%F A059117 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.
%e A059117 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.
%t A059117 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 *)
%Y A059117 Sum of rows gives A055203. Columns include A000007, A057427, A058809, A059116. Final positive number in each row is A000680.
%K A059117 nonn,tabl
%O A059117 0,18
%A A059117 _Henry Bottomley_, Jan 05 2001