0,3

The formula for the PDGF4(z;n) polynomials in the GF4 denominators of A156933 can be found below.

The general structure of the GFKT4(z;p) that generate the z^p coefficients of the PDGF4(z;n) polynomials can also be found below. The KT4(z;p) polynomials in the nominators of the GFKT4(z;p) have a nice symmetrical structure.

The sequence of the number of terms of the first few KT4z;p) polynomials is: 1, 3, 5, 7, 10, 13, 15, 18, 20, 23, 26, 29, 32, 34, 37, 40, 42. The differences of this sequence and that of the number of terms of the KT3(z;p), see A157704, follow a simple pattern.

A Maple algorithm that generates relevant GFKT4(z;p) information can be found below.

PDGF4(z;n) = product((1-(2*n+1-2*k)*z)^(3*k+1), k=0..n) with n = 1, 2, 3, ..

GFKT4(z;p) = (-1)^(p)*(z^q4)*KT4(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, .. .

Some PDGF4 (z;n) are:

PDGF4(z; n=3) = (1-7*z)*(1-5*z)^4*(1-3*z)^7*(1-z)^10

PDGF4(z; n=4) = (1-9*z)*(1-7*z)^4*(1-5*z)^7*(1-3*z)^10*(1-z)^13

The first few GFKT4's are:

GFKT4(z;p=0) = 1/(1-z)

GFKT4(z;p=1) = -(1+3*z+2*z^2)/(1-z)^4

GFKT4(z;p=2) = z*(18+128*z+171*z^2+42*z^3+z^4)/(1-z)^7

Some KT4(z,p) polynomials are:

KT4(z;p=2) = 18+128*z+171*z^2+42*z^3+z^4

KT4(z;p=3) = 22+1348*z+11738*z^2+26293*z^3+17693*z^4+3271*z^5+115*z^6

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*n2+1-(2*k))*z)^(3*k+1), 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)); GFKT4(p):=sum((fk)*z^k, k=0..infinity); q4:=ldegree((numer (GFKT4(p)))): KT4(p):=sort((-1)^(p)*simplify((GFKT4(p)*(1-z)^(3*p+1))/z^q4), z, ascending);

Originator sequence A156933

See A081436 for the z^1 coefficients and A157708 for the z^2 coefficients.

Row sums equal A064350 and those of A157704

Cf. A157702, A157703, A157704

Sequence in context: A026345 A092644 A006281 this_sequence A078073 A075568 A057026

Adjacent sequences: A157702 A157703 A157704 this_sequence A157706 A157707 A157708

easy,nonn,tabf

Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009