

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
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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)/(1z)^(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(nk), k=1 .. 3*p+1) with p = 0, 1, 2, .. .


EXAMPLE

Some PDGF3 (z;n) are:
PDGF3(z;n=3) = (1z)*(12*z)^4*(13*z)^7*(14*z)^10
PDGF3(z;n=4) = (1z)*(12*z)^4*(13*z)^7*(14*z)^10*(15*z)^13
The first few GFKT3's are:
GFKT3(z;p=0) = 1/(1z)
GFKT3(z;p=1) = (5*z+1)/(1z)^4
GFKT3(z;p=2) = z*(32+186*z+132*z^2+10*z^3)/(1z)^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(nn1), 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)*(1z)^(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



