A196346 -id:A196346 - cp's OEIS frontend

A196345 The 5th power of the g.f. equals the g.f. of A196344.

1, 1, -2, 6, -21, 80, -320, 1326, -5637, 24434, -107542, 479196, -2157045, 9792702, -44780606, 206055345, -953305628, 4431463845, -20686696836, 96931500441, -455722376856, 2149086834269, -10162544424168, 48176923110789, -228913128188293, 1089973053510359

Offset: 0

Paul D. Hanna, Oct 01 2011

sign

A196344 is defined as the coefficients in the g.f. Q(x), where -2 <= A196344(n) <= 2 for all n>1 with initial terms {1,5}, such that Q(x)^(1/5) consists entirely of integer coefficients.

Limit a(n+1)/a(n) = -5.0015989761 6639938823 4051883169 0463138590 3476719792 3351242105 ...

			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 +...
where
A(x)^5 = 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 +...+ A196344(n)*x^n +...
A196344 begins: [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,...].

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

Cf. A196344, A196346, A106223.

{a(n)=local(A=1+5*x); if(n==0, 1, 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))^(1/5), 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))}

cp's OEIS Frontend

A196345 The 5th power of the g.f. equals the g.f. of A196344.

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