OFFSET
1,3
COMMENTS
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).
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.
FORMULA
E.g.f. for the row polynomials, which are a(n, t) for n > 0, is:
(t-1) / (t - exp(x*(t-1)/(1-x*(t-1)))).
E.g.f. for the polynomials b(n, t), introduced above, is the reciprocal of that.
MATHEMATICA
CoefficientList[#, t] & /@ (# Range@Length@#!) &@ Rest@CoefficientList[(t-1) / (t - Exp[x(t-1)/(1-x(t-1))]) + O[x]^10 // Simplify, x] // Flatten (* Andrey Zabolotskiy, Feb 19 2024 *)
PROG
(PARI) T(n) = [Vecrev(p) | p<-Vec(-1 + serlaplace((y-1) / (y - exp(x*(y-1)/(1-x*(y-1)) + O(x*x^n) ))))]
{ my(A=T(7)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 19 2024
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Tom Copeland, Oct 22 2007, Nov 30 2007
EXTENSIONS
Rows 7-9 added and offset changed by Andrey Zabolotskiy, Feb 19 2024
STATUS
approved