A131202 A coefficient tree from the list partition transform relating A111884, A084358, A000262, A094587, A128229 and A131758.

1, -1, 3, 1, -8, 13, 1, 11, -61, 73, -19, 66, 66, -494, 501, 151, -993, 2102, -298, -4293, 4051, -1091, 9528, -33249, 52816, -21069, -39528, 37633, 7841, -82857, 378261, -929101, 1207299, -560187, -375289, 394353, -56519, 692422, -3832928, 12255802, -23834210, 26643994, -12620672, -3481562, 4596553

Offset: 1

Author

Tom Copeland, Oct 22 2007, Nov 30 2007

Keywords

• sign
• tabl

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).

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.

Programs

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 *)

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

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.

Extensions

Rows 7-9 added and offset changed by Andrey Zabolotskiy, Feb 19 2024