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A083350 Integer coefficients of a power series A(x) such that A(x)^3 = A083349(x). 4

%I

%S 1,1,1,-1,3,0,-6,17,-17,-19,114,-215,111,609,-2084,2947,1187,-16252,

%T 38872,-32709,-87431,390618,-673709,47692,3018098,-8616766,9761812,

%U 13605710,-84546525,171930010,-77194029,-610108400,2090199824,-2940478260,-1840404119,19501756943,-46202080484

%N Integer coefficients of a power series A(x) such that A(x)^3 = A083349(x).

%C Self-convolution cube equals A083349.

%C A083349 is the minimal permutation of the positive integers having a self-convolution cube-root consisting entirely of integers.

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%e A083349(x)^(1/3) = A(x) = 1 + x + x^2 - x^3 + 3x^4 + 0x^5 - 6x^6 + ...

%t n = 40; A = 1 + 3x; P = Table[0, 3(n+1)]; P[[1]] = 1; P[[3]] = 2; For[j = 2, j <= n, j++, For[k = 2, k <= 3(n+1), k++, If[P[[k]] == 0, t = Coefficient[(A + k x^j + x^2 O[x]^j)^(1/3), x, j]; If[Denominator[t] == 1, P[[k]] = j+1; A = A + k x^j; Break[]]]]];

%t CoefficientList[A^(1/3) + O[x]^n, x] (* _Jean-Fran├žois Alcover_, Jul 26 2018, from PARI *)

%o (PARI) {a(n)=local(A=1+3*x,P=vector(3*(n+1)));P[1]=1;P[3]=2; for(j=2,n, for(k=2,3*(n+1),if(P[k]==0, t=polcoeff((A+k*x^j+x^2*O(x^j))^(1/3),j); if(denominator(t)==1,P[k]=j+1;A=A+k*x^j;break)))); return(polcoeff((A+x*O(x^n))^(1/3),n))}

%Y Cf. A083349, A106213, A106214.

%K sign

%O 0,5

%A _Paul D. Hanna_, Apr 25 2003; revised May 01 2005

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Last modified January 20 16:04 EST 2022. Contains 350472 sequences. (Running on oeis4.)