A196345 -id:A196345 - cp's OEIS frontend

A352703 G.f. A(x) satisfies: A(x)^5 = A(x^5) + 5*x.

1, 1, -2, 6, -21, 80, -320, 1326, -5637, 24434, -107542, 479196, -2157045, 9792702, -44780606, 206055346, -953305632, 4431463863, -20686696920, 96931500840, -455722378776, 2149086843549, -10162544469252, 48176923330632, -228913129263389, 1089973058779915

Offset: 0

Paul D. Hanna, Mar 29 2022

sign

Not the same as A106223 or A196345.

			G.f.: A(x) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 + 479196*x^11 + ...
such that A(x)^5 = A(x^5) + 5*x, as illustrated by:
A(x)^5 = 1 + 5*x + x^5 - 2*x^10 + 6*x^15 - 21*x^20 + 80*x^25 - 320*x^30 + 1326*x^35 - 5637*x^40 + 24434*x^45 - 107542*x^50 + ...

Cf. A107092, A352704, A352705.

{a(n) = my(A=1+x); for(i=0,n\5,
A = (subst(A,x,x^5) + 5*x + x*O(x^(5*n)))^(1/5));
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A196344 Coefficients of g.f. A(x) where -2 <= a(n) <= 2 for all n>1, with initial terms {1,5}, such that A(x)^(1/5) consists entirely of integer coefficients.

Original entry on oeis.org

1, 5, 0, 0, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Oct 01 2011

sign

A(z) = 0 at z = -0.1999360614 0060333411 6048445601 7524456163 6606200034 1004259693 ...

			G.f.: A(x) = 1 + 5*x + x^5 - 2*x^10 + x^15 - x^20 + x^35 - 2*x^40 - x^45 - 2*x^50 + x^55 + 2*x^65 - x^70 + 2*x^80 +...
where the following series consists entirely of integer coefficients:
A(x)^(1/5) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 +...+ A196345(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A196345 (5th root), A196346 (quintisection), A106222 (variant).

{a(n)=local(A=1+5*x); if(n==0, 1, if(n%5==0,for(j=1, n, for(k=-2, 2, t=polcoeff((A+k*x^j+x*O(x^j))^(1/5), j);
if(denominator(t)==1, A=A+k*x^j; break)))); polcoeff(A+x*O(x^n), n))}

A196346 A quintisection of A196344.

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 0, 1, -2, -1, -2, 1, 0, 2, -1, 0, 2, 0, -1, 1, -1, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, 0, -2, 0, 0, -2, 2, -2, -1, 0, -1, 2, -1, -2, 1, -2, 1, 2, -2, 0, 2, 2, 1, 0, -2, -1, -2, -2, 0, 2, 2, 1, 1, 1, 2, 1, -1, 2, 1, -1, 1, 0, 2, 2, -1, 2, -1, 1, -1, -2, -1, 1, -1, -1, -1, 1, 2, 0, -2, 0, 1, 2, 1, 2, -2

Offset: 0

Paul D. Hanna, Oct 01 2011

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			G.f.: A(x) = 1 + x - 2*x^2 + x^3 - x^4 + x^7 - 2*x^8 - x^9 - 2*x^10 + x^11 + 2*x^13 - x^14 + 2*x^16 - x^18 + x^19 - x^20 +...
where the following series consists entirely of integer coefficients:
(A(x^5) + 5*x)^(1/5) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 +...+ A196345(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A196344, A196345.

{a(n)=local(A=1+5*x); if(n==0, 1, for(j=1, 5*n, for(k=-2, 2, t=polcoeff((A+k*x^j+x*O(x^j))^(1/5), j);
if(denominator(t)==1, A=A+k*x^j; break))); polcoeff(A+x*O(x^(5*n)), 5*n))}

cp's OEIS Frontend

A352703 G.f. A(x) satisfies: A(x)^5 = A(x^5) + 5*x.

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A196344 Coefficients of g.f. A(x) where -2 <= a(n) <= 2 for all n>1, with initial terms {1,5}, such that A(x)^(1/5) consists entirely of integer coefficients.

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A196346 A quintisection of A196344.

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