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%I #27 Feb 20 2024 01:12:16
%S 1,-1,3,1,-8,13,1,11,-61,73,-19,66,66,-494,501,151,-993,2102,-298,
%T -4293,4051,-1091,9528,-33249,52816,-21069,-39528,37633,7841,-82857,
%U 378261,-929101,1207299,-560187,-375289,394353,-56519,692422,-3832928,12255802,-23834210,26643994,-12620672,-3481562,4596553
%N A coefficient tree from the list partition transform relating A111884, A084358, A000262, A094587, A128229 and A131758.
%C Construct the infinite array of polynomials
%C a(0,t) = 1
%C a(1,t) = 1
%C a(2,t) = -1 + 3*t
%C a(3,t) = 1 - 8*t + 13*t^2
%C a(4,t) = 1 + 11*t - 61*t^2 + 73*t^3
%C a(5,t) = -19 + 66*t + 66*t^2 - 494*t^3 + 501*t^4
%C a(6,t) = 151 - 993*t + 2102*t^2 - 298*t^3 - 4293*t^4 + 4051*t^5
%C 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.
%C b(0,t) = 1 and b(n,t) = -A000262(n)*(t-1)^(n-1) for n > 0.
%C Then A111884(n) = a(n,0).
%C Lower triangular matrix A094587 = binomial(n,k)*a(n-k,1).
%C A084358(n) = a(n,2).
%C Signed A128229 = matrix inverse of binomial(n,k)*a(n-k,1) = binomial(n,k)*b(n-k,1) = A132013.
%C As t tends to infinity, a(n,t)/t^(n-1) tends to A000262(n) for n > 0.
%C 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.
%F E.g.f. for the row polynomials, which are a(n, t) for n > 0, is:
%F (t-1) / (t - exp(x*(t-1)/(1-x*(t-1)))).
%F E.g.f. for the polynomials b(n, t), introduced above, is the reciprocal of that.
%t 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 *)
%o (PARI) T(n) = [Vecrev(p) | p<-Vec(-1 + serlaplace((y-1) / (y - exp(x*(y-1)/(1-x*(y-1)) + O(x*x^n) ))))]
%o { my(A=T(7)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Feb 19 2024
%K sign,tabl
%O 1,3
%A _Tom Copeland_, Oct 22 2007, Nov 30 2007
%E Rows 7-9 added and offset changed by _Andrey Zabolotskiy_, Feb 19 2024
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