A131202 - OEIS

%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