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A157702 G.f.s of the z^p coefficients of the polynomials in the GF1 denominators of A156921 7
1, 1, 1, 7, 26, 7, 3, 166, 951, 951, 166, 3, 263, 8999, 59637, 108602, 59637, 8999, 263, 174, 33124, 848555, 6062651, 15477896, 15477896, 6062651, 848555, 33124, 174, 45, 66963, 5856626, 122966782, 920090513 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The formula for the PDGF1(z;n) polynomials in the GF1 denominators of A156921 can be found below.

The general structure of the GFKT1(z;p) that generate the z^p coefficients of the PDGF1(z; n) polynomials can also be found below. The KT1(z;p) polynomials in the nominators of the GFKT1(z;p) have a nice symmetrical structure.

The sequence of the number of terms of the first few KT1(z;p) polynomials is: 1, 2, 3, 6, 7, 10, 13, 14, 17, 20, 23, 24, 27, 30, 33, 36, 37, 40. The first differences follow a simple pattern. The positions of the 1's follow the Lazy Caterer's sequence A000124.

A Maple algorithm that generates relevant GFKT1(z;p) information can be found below.

LINKS

Table of n, a(n) for n=0..33.

FORMULA

PDGF1(z;n) = product( (1-(2*m-1)*z)^(n+1-m) , m=1..n) with n = 1, 2, 3, .. .

GFKT1(z;p) = (-1)^(p)*(z^q1)*KT1(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, .. .

The recurrence relation for the z^p coefficients a(n) is: a(n) = sum((-1)^(k+1)* binomial(3*p + 1, k)*a(n-k), k=1 .. 3*p+1) with p = 0, 1, 2, .. .

EXAMPLE

Some PDGF1 (z;n) are:

PDGF1(z;n=3) = (1-5*z)*(1-3*z)^2*(1-z)^3

PDGF1(z;n=4) = ((1-7*z)*(1-5*z)^2*(1-3*z)^3*(1-z)^4)

The first few GFKT1's are:

GFKT1(z;p=0) = 1/(1-z)

GFKT1(z;p=1) = -z*(1+z)/(1-z)^4

GFKT1(z;p=2) = z^2*(7+26*z+7*z^2)/(1-z)^7

Some KT1(z;p) polynomials are:

KT1(z;p=2) = 7+26*z+7*z^2

KT1(z;p=3) = 3+166*z+951*z^2+951*z^3+166*z^4+3*z^5

KT1(z;p=4) = 263+8999*z+59637*z^2+108602*z^3+59637*z^4+8999*z^5+263*z^6

MAPLE

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1, n1) *a(n-n1), n1=1..3*p+1): fk:=rsolve(a(n) = fn, a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(2*m-1)*z)^(n2+1-m), m=1..n2); a(n2):= coeff(fz(n2), z, p); end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT1(p):=(sum(fk*z^k, k=0..infinity)); q1:=ldegree((numer(GFKT1(p)))): KT1(p):=sort((-1)^p*simplify((GFKT1(p))*(1-z)^(3*p+1)/z^q1), z, ascending);

CROSSREFS

Originator sequence A156921

See A000330 for the z^1 coefficients and A157706 for the z^2 coefficients.

Row sums equal A052502

Cf. A157703, A157704, A157705

Sequence in context: A081298 A214593 A012490 * A063453 A284054 A262109

Adjacent sequences:  A157699 A157700 A157701 * A157703 A157704 A157705

KEYWORD

easy,nonn,tabf

AUTHOR

Johannes W. Meijer, Mar 07 2009

STATUS

approved

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Last modified May 28 21:38 EDT 2017. Contains 287241 sequences.