A084205 -id:A084205 - cp's OEIS frontend

A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

1, 1, -1, 3, -7, 20, -58, 177, -554, 1769, -5739, 18866, -62684, 210146, -709882, 2413743, -8253995, 28366316, -97916761, 339326189, -1180068800, 4116957243, -14404398636, 50530280752, -177684095927, 626181400993, -2211215950469, 7823025701314, -27724997048327

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.269562488839799 where A(r)=0.

Robert Israel, Table of n, a(n) for n = 0..1000
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. A083954, A084202, A084203, A084205-A084212.

g:= 1: a[0]:= 1:
for n from 1 to 50 do
  a[n]:= -floor((coeff(g^4,x,n)-1)/4);
  g:= g + a[n]*x^n;
od:
seq(a[n],n=0..50); # Robert Israel, Sep 04 2019

kmax = 30;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^4 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1}; coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 4, 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 *)

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.

Original entry on oeis.org

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 *)

A110631 Every 5th term of A083945 such that the self-convolution 5th power is congruent modulo 25 to A083945, which consists entirely of numbers 1 through 5.

Original entry on oeis.org

1, 1, 4, 3, 2, 4, 4, 2, 1, 5, 2, 1, 5, 1, 3, 2, 5, 3, 4, 4, 5, 4, 5, 2, 1, 5, 4, 1, 2, 5, 1, 5, 1, 1, 1, 2, 3, 4, 2, 2, 4, 3, 2, 5, 2, 3, 5, 1, 1, 2, 3, 3, 1, 1, 2, 2, 3, 4, 4, 1, 2, 1, 3, 4, 1, 4, 2, 3, 5, 4, 4, 3, 5, 3, 4, 2, 2, 4, 2, 2, 5, 3, 2, 4, 2, 5, 5, 5, 3, 5, 4, 4, 1, 3, 5, 1, 5, 5, 4, 3, 5, 2, 2, 2, 5

Offset: 0

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

nonn

Congruent modulo 5 to A084205, where the self-convolution 5th power of A084205 equals A083945.

Cf. A083945, A111583, A084205.

{a(n)=local(p=5,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) = A083945(5*n) for n>=0.
G.f. satisfies: A(x^5) = G(x) - 5*x*((1-x^4)/(1-x))/(1-x^5), where G(x) is the g.f. of A083945.
G.f. satisfies: A(x)^5 = A(x^5) + 5*x*((1-x^4)/(1-x))/(1-x^5) + 25*x^2*H(x) where H(x) is the g.f. of A111583.

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A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

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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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A110631 Every 5th term of A083945 such that the self-convolution 5th power is congruent modulo 25 to A083945, which consists entirely of numbers 1 through 5.

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