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A131202 A coefficient tree from the list partition transform relating A111884, A084358, A000262, A094587, A128229 and A131758. 2
1, -1, 3, 1, -8, 13, 1, 11, -61, 73, -19, 66, 66, -494, 501, 151, -993, 2102, -298, -4293, 4051 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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) .

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] .

LINKS

Table of n, a(n) for n=0..20.

CROSSREFS

Sequence in context: A049760 A019146 A102537 * A067955 A182509 A049965

Adjacent sequences:  A131199 A131200 A131201 * A131203 A131204 A131205

KEYWORD

easy,sign

AUTHOR

Tom Copeland, Oct 22 2007, Nov 30 2007

STATUS

approved

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Last modified December 22 14:55 EST 2014. Contains 252364 sequences.