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

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

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: A193783 A271398 A006214 * A270565 A271165 A015541

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 | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 6 03:02 EST 2016. Contains 278771 sequences.