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A173410
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The polynomial array L_q(n,k), evaluated at q = -1, i.e., L_{-1}(n,k).
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0
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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, 64, 48, 56, 24, 8, 1
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OFFSET
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1,3
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COMMENTS
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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}.
Sequence above is L_q(n,k) evaluated at q = -1 (the nonzero values).
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).
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REFERENCES
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J. Lindsay, T. Mansour, and M. Shattuck, Generalizing a Relation between Polynomial Bases, Tech. Report, 2010.
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LINKS
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FORMULA
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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.
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)
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].
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).
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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