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A173410 The polynomial array L_q(n,k), evaluated at q = -1, i.e., L_{-1}(n,k). 0


%S 1,1,2,1,2,2,1,4,4,4,1,4,4,8,4,1,8,8,20,12,6,1,8,8,28,20,18,6,1,16,16,

%T 64,48,56,24,8,1

%N The polynomial array L_q(n,k), evaluated at q = -1, i.e., L_{-1}(n,k).

%C L_q(n,k) is a q-generalization of the Lah number L(n,k) (see A105278) and is given by the two-term recurrence L_q(n,k) = L_q(n-1,k-1) + [(n-1)_q + k_q]*L_q(n-1,k), for all positive integers n and k, with the boundary values L_q(n,0) = delta_{n,0} and L_q(0,k) = delta_{k,0} for all nonnegative integers n and k, where m_q:=1+q+...+q^{m-1}.

%C Sequence above is L_q(n,k) evaluated at q = -1 (the nonzero values).

%C L_q(n,k) also arises as a distribution polynomial for a certain inversion statistic defined on the set of Lah distributions enumerated by L(n,k).

%D J. Lindsay, T. Mansour, and M. Shattuck, Generalizing a Relation between Polynomial Bases, Tech. Report, 2010.

%H Jim Lindsay, Toufik Mansour and Mark Shattuck, <a href="http://dx.doi.org/10.4310/JOC.2011.v2.n2.a4">A new combinatorial interpretation of a g-analogue of the Lah numbers</a>, Journal of Combinatorics, Volume 2, Number 2, 245-264, 2011.

%F Recurrence: The L_{-1}(n,k) are generated by the two-term recurrence L_{-1}(n,k)= L_{-1}(n-1,k-1) + [(n-1)_{-1}+k_{-1}]*L_{-1}(n-1,k), for all positive integers n and k, with the boundary conditions as above for L_q(n,k), and where m_{-1} = [m is odd] for m a nonnegative integer.

%F Generating Function: 1 + sum_{n>=1} sum_{1 <= k <= n} L_{-1}(n,k)*x^n*y^k = (1 - 2x^2 - x^2y^2 + xy + 2x^3y^2 + 2x^2y - x^3y^3)/(1 - 2x^2 -2x^2y^2 - 2x^4y^2 + x^4y^4)

%F Connection Constant Relation: The L_{-1}(n,k) are also uniquely determined by the polynomial relations x*(x+1_{-1})*(x+2_{-1})*...*(x+(n-1)_{-1})= sum_{k=1}^n L_{-1}(n,k)*x*(x-1_{-1})*...*(x-(k-1)_{-1}), for positive integers n, where m_{-1} = [m is odd].

%F Row Sum: sum_{k=0}^n L_{-1}(n,k) = f_r, where r = [3*n/2] and f_m denotes the Fibonacci sequence given by the recurrence f_m=f_{m-1}+f_{m-2} if m >= 2, with f_0=f_1=1 (see A000045).

%Y See also A105278 and A000045.

%K nonn

%O 1,3

%A _Mark Shattuck_, Feb 17 2010

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