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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

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Last modified September 21 15:26 EDT 2014. Contains 247025 sequences.