0,3

The formula for the PDGF2(z;n) polynomials in the GF2 denominators of A156925 can be found below.

The general structure of the GFKT2(z;p) that generate the z^p coefficients of the PDGF2(z; n) polynomials can also be found below. The KT2(z;p) polynomials in the nominators of the GFKT2(z;p) have a nice symmetrical structure.

The sequence of the number of terms of the first few KT2(z;p) polynomials is: 1, 1, 2, 5, 6, 9, 12, 13, 16, 19, 22, 23, 26. The first differences follow a simple pattern. The positions of the 1's follow the Lazy Caterer's sequence A000124 with one exception, here a(0) = 0.

A Maple algorithm that generates relevant GFKT2(z;p) information can be found below.

PDGF2(z;n) = product((1-m*z)^(n+1-m),m=1..n) with n = 1, 2, 3, ..

GFKT2(z;p) = (-1)^(p)*(z^q2)*KT2(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 PDGF2 (z;n) are:

PDGF2(z;n=3) = (1-z)^3*(1-2*z)^2*(1-3*z)

PDGF2(z;n=4) = (1-z)^4*(1-2*z)^3*(1-3*z)^2*(1-4*z)

The first few GFKT2's are:

GFKT2(z;p=0) = 1/(1-z)

GFKT2(z;p=1) = -z/(z-1)^4

GFKT2(z;p=2) = z^2*(5+5*z)/(1-z)^7

Some KT2(z,p) polynomials are:

KT2(z;p=2) = 5+5*z

KT2(z;p=3) = 2+62*z+152*z^2+62*z^3+2*z^4

KT2(z;p=4) = 91+1652*z+5957*z^2+5957*z^3+1652*z^4+91*z^5

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-m*z)^(n2+1-m), m=1..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)); GFKT2(p):=sum((fk)*z^k, k=0..infinity); q2:=ldegree((numer(GFKT2(p)))): KT2(p):=sort((-1)^p*simplify((GFKT2(p)*(1-z)^(3*p+1))/z^q2), z, ascending);

Originator sequence A156925

See A000292 for the z^1 coefficients and A040977 for the z^2 coefficients divided by 5.

Row sums equal A025035

Cf. A157702, A157704, A157705

Adjacent sequences: A157700 A157701 A157702 this_sequence A157704 A157705 A157706

Sequence in context: A023580 A021648 A128006 this_sequence A158349 A079384 A158274

easy,nonn,tabf

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