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COMMENT
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Construct the infinite array of polynomials
a(0,t) = 1
a(1,t) = 1
a(2,t) = -1 + 3 t
a(3,t) = 1 - 8 t + 13 t^2
a(4,t) = 1 + 11 t - 61 t^2 + 73 t^3
a(5,t) = -19 + 66 t + 66 t^2 - 494 t^3 + 501 t^4
a(6,t) =151 - 993 t + 2102 t^2 - 298 t^3 - 4293 t^4 + 4051 t^5
This array is the reciprocal array of the following array b(n,t) under the list partition transform and its associated operations described in A133314.
b(0,t) = 1 and b(n,t) = -A000262(n)*(t-1)^(n-1) for n>0 .
Then, A111884(n) = a(n,0) .
Lower triangular matrix A094587 = binomial(n,k)*a(n-k,1) .
A084358(n) = a(n,2) .
Signed A128229 = matrix inverse of binomial(n,k)*a(n-k,1) = binomial(n,k)*b(n-k,1) = A132013 .
As t tends to infinity, a(n,t)/t^(n-1) tends to A000262(n) for n>0.
The P(n,t) of A131758 can be constructed from T(n,k,t) = binomial(n,k)*a(n-k,t) by letting T(n,k,t) multiply the column vector c(n,t) given by c(0,t) = 0! and c(n,t) = n! (t-1)^(n-1) for n>0. The P(n,t) have rich associations to other sequences.
exp[b(.,t)*x] = { t - exp[ x*(t-1) / [1-x*(t-1)] ] } / (t-1) and exp[a(.,t)*x] = 1 / exp[b(.,t)*x] .
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