login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927 8
1, 1, 5, 32, 186, 132, 10, 56, 2814, 17834, 27324, 11364, 1078, 10, 48, 17988, 494720, 3324209, 7526484, 6382271, 2004296, 203799, 4580, 5, 16, 72210, 7108338, 146595355, 1025458635, 2957655028, 3828236468 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The formula for the PDGF3(z;n) polynomials in the GF3 denominators of A156927 can be found below.

The general structure of the GFKT3(z;p) that generate the z^p coefficients of the PDGF3(z; n) polynomials can also be found below. The KT3(z;p) polynomials in the nominators of the GFKT3(z; p) have a nice symmetrical structure.

The sequence of the number of terms of the first few KT3(z;p) polynomials is: 1, 2, 4, 7, 10, 13, 14, 17, 20, 23, 26, 29, 32, 34, 36, 39, 42. The differences of this sequence and that of the number of terms of the KT4(z;p), see A157705, follow a simple pattern.

A Maple algorithm that generates relevant GFKT3(z;p) information can be found below.

LINKS

Table of n, a(n) for n=0..30.

FORMULA

PDGF3(z;n) = product((1-(k+1)*z)^(1+3*k), k=0..n) with n = 1, 2, 3, ..

GFKT3(z;p) = (-1)^(p)*(z^q3)*KT3(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 PDGF3 (z;n) are:

PDGF3(z;n=3) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10

PDGF3(z;n=4) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10*(1-5*z)^13

The first few GFKT3's are:

GFKT3(z;p=0) = 1/(1-z)

GFKT3(z;p=1) = -(5*z+1)/(1-z)^4

GFKT3(z;p=2) = z*(32+186*z+132*z^2+10*z^3)/(1-z)^7

Some KT3(z,p) polynomials are:

KT3(z;p=2) = 32+186*z+132*z^2+10*z^3

KT3(z;p=3) = 56+2814*z+17834*z^2+27324*z^3+11364*z^4+1078*z^5+10*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-(k+1)*z)^(1+3*k), k=0..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)); GFKT3(p):=sum((fk)*z^k, k=0..infinity); q3:=ldegree((numer(GFKT3(p)))): KT3(p):=sort((-1)^(p)*simplify((GFKT3(p)*(1-z)^(3*p+1))/z^q3), z, ascending);

CROSSREFS

Originator sequence A156927

See A002414 for the z^1 coefficients and A157707 for the z^2 coefficients divided by 2.

Row sums equal A064350 and those of A157705

Cf. A157702, A157703, A157705

Sequence in context: A177467 A193783 A006214 * A015541 A024064 A164594

Adjacent sequences:  A157701 A157702 A157703 * A157705 A157706 A157707

KEYWORD

easy,nonn,tabf

AUTHOR

Johannes W. Meijer, Mar 07 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 21 17:12 EST 2014. Contains 252324 sequences.