A157702 G.f.s of the z^p coefficients of the polynomials in the GF1 denominators of A156921.

1, 1, 1, 7, 26, 7, 3, 166, 951, 951, 166, 3, 263, 8999, 59637, 108602, 59637, 8999, 263, 174, 33124, 848555, 6062651, 15477896, 15477896, 6062651, 848555, 33124, 174, 45, 66963, 5856626, 122966782, 920090513

Offset: 0

Author

Johannes W. Meijer, Mar 07 2009

Keywords

• easy
• nonn
• tabf
• uned

Comments

The formula for the PDGF1(z;n) polynomials in the GF1 denominators of A156921 can be found below.

The general structure of the GFKT1(z;p) that generate the z^p coefficients of the PDGF1(z; n) polynomials can also be found below. The KT1(z;p) polynomials in the numerators of the GFKT1(z;p) have a nice symmetrical structure.

The sequence of the number of terms of the first few KT1(z;p) polynomials is: 1, 2, 3, 6, 7, 10, 13, 14, 17, 20, 23, 24, 27, 30, 33, 36, 37, 40. The first differences follow a simple pattern. The positions of the 1's follow the Lazy Caterer's sequence A000124.

A Maple algorithm that generates relevant GFKT1(z;p) information can be found below.

Examples

			Some PDGF1 (z;n) are:
  PDGF1(z;n=3) = (1-5*z)*(1-3*z)^2*(1-z)^3
  PDGF1(z;n=4) = ((1-7*z)*(1-5*z)^2*(1-3*z)^3*(1-z)^4)
The first few GFKT1's are:
  GFKT1(z;p=0) = 1/(1-z)
  GFKT1(z;p=1) = -z*(1+z)/(1-z)^4
  GFKT1(z;p=2) = z^2*(7+26*z+7*z^2)/(1-z)^7
Some KT1(z;p) polynomials are:
  KT1(z;p=2) = 7+26*z+7*z^2
  KT1(z;p=3) = 3+166*z+951*z^2+951*z^3+166*z^4+3*z^5
  KT1(z;p=4) = 263+8999*z+59637*z^2+108602*z^3+59637*z^4+8999*z^5+263*z^6
		

Crossrefs

Originator sequence A156921.

See A000330 for the z^1 coefficients and A157706 for the z^2 coefficients.

Row sums equal A052502.

Cf. A157703, 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-(2*m-1)*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)); GFKT1(p):=(sum(fk*z^k,k=0..infinity)); q1:=ldegree((numer(GFKT1(p)))): KT1(p):=sort((-1)^p*simplify((GFKT1(p))*(1-z)^(3*p+1)/z^q1),z, ascending);

Formula

PDGF1(z;n) = Product_{m=1..n} (1-(2*m-1)*z)^(n+1-m) with n = 1, 2, 3, ... .

GFKT1(z;p) = (-1)^(p)*(z^q1)*KT1(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, ... .