A157703 G.f.s of the z^p coefficients of the polynomials in the GF2 denominators of A156925.

1, 1, 5, 5, 2, 62, 152, 62, 2, 91, 1652, 5957, 5957, 1652, 91, 52, 5240, 77630, 342188, 551180, 342188, 77630, 5240, 52, 12, 8549, 424921, 5629615, 28123559, 61108544, 61108544, 28123559, 5629615, 424921, 8549, 12

Offset: 0

Author

Johannes W. Meijer, Mar 07 2009

Keywords

• easy
• nonn
• tabf
• uned

Comments

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 numerators 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.

Examples

			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
		

Crossrefs

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.

Programs

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-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);

Formula

PDGF2(z;n) = Product_{m=1..n} (1-m*z)^(n+1-m) 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_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .