A084207 -id:A084207 - cp's OEIS frontend

A084206 G.f. A(x) defined by: A(x)^6 consists entirely of integer coefficients between 1 and 6 (A083946); A(x) is the unique power series solution with A(0)=1.

1, 1, -2, 7, -27, 115, -510, 2343, -11029, 52896, -257457, 1268098, -6307546, 31633044, -159757597, 811708539, -4145882814, 21273287952, -109603172373, 566748274099, -2940175511195, 15297961574259, -79808998488751, 417373462315834

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.1815238859919 where A(r)=0.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083946, A084202-A084205, A084207-A084212.

kmax = 25;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^6 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 6, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084208 G.f. A(x) defined by: A(x)^8 consists entirely of integer coefficients between 1 and 8 (A083948); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -3, 15, -82, 484, -2992, 19110, -124979, 832234, -5621028, 38402783, -264858143, 1841221687, -12886279885, 90713376563, -641815393278, 4561172770669, -32542369727538, 232992967457839

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.131401689761435 where A(r)=0.

Cf. A083948, A084202-A084207, A084209-A084212.

kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^8 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 8, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A110635 Every 7th term of A083947 such that the self-convolution 7th power is congruent modulo 49 to A083947, which consists entirely of numbers 1 through 7.

Original entry on oeis.org

1, 1, 5, 1, 1, 4, 2, 1, 1, 3, 5, 1, 2, 5, 1, 7, 6, 4, 4, 6, 4, 5, 7, 3, 4, 2, 4, 3, 3, 2, 7, 4, 6, 6, 3, 1, 1, 6, 5, 6, 6, 3, 1, 2, 5, 7, 3, 3, 7, 5, 5, 6, 4, 6, 3, 4, 2, 5, 4, 4, 7, 3, 4, 1, 5, 6, 7, 2, 2, 5, 4, 1, 4, 4, 1, 1, 4, 3, 6, 7, 6, 2, 6, 6, 2, 1, 6, 6, 1, 5, 2, 2, 5, 5, 4, 2, 3, 7, 4, 5, 1, 3, 6, 4, 4

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 08 2005

nonn

Congruent modulo 7 to A084207, where the self-convolution 7th power of A084207 equals A083947.

Cf. A083947, A111584, A084207.

{a(n)=local(p=7,A,C,X=x+x*O(x^(p*n)));if(n==0,1, A=sum(i=0,n-1,a(i)*x^(p*i))+p*x*((1-x^(p-1))/(1-X))/(1-X^p); for(k=1,p,C=polcoeff((A+k*x^(p*n))^(1/p),p*n); if(denominator(C)==1,return(k);break)))}

a(n) = A083947(7*n) for n>=0.
G.f. satisfies: A(x^7) = G(x) - 7*x*((1-x^6)/(1-x))/(1-x^7), where G(x) is the g.f. of A083947.
G.f. satisfies: A(x)^7 = A(x^7) + 7*x*((1-x^6)/(1-x))/(1-x^7) + 49*x^2*H(x) where H(x) is the g.f. of A111584.

cp's OEIS Frontend

A084206 G.f. A(x) defined by: A(x)^6 consists entirely of integer coefficients between 1 and 6 (A083946); A(x) is the unique power series solution with A(0)=1.

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A084208 G.f. A(x) defined by: A(x)^8 consists entirely of integer coefficients between 1 and 8 (A083948); A(x) is the unique power series solution with A(0)=1.

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A110635 Every 7th term of A083947 such that the self-convolution 7th power is congruent modulo 49 to A083947, which consists entirely of numbers 1 through 7.

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