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A157703
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G.f.s of the z^p coefficients of the polynomials in the GF2 denominators of A156925.
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7
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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, ... .
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EXAMPLE
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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
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MAPLE
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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);
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CROSSREFS
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See A000292 for the z^1 coefficients and A040977 for the z^2 coefficients divided by 5.
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KEYWORD
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easy,nonn,tabf,uned
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AUTHOR
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STATUS
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approved
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