A357163 - OEIS

%I #9 Sep 20 2022 21:21:33

%S 1,1,5,38,313,2834,27088,269380,2757797,28872568,307696566,3326835855,

%T 36403128996,402370063992,4485931975701,50386112677647,

%U 569624341701738,6476615022560400,74013180802610161,849642206122063571,9793310961240979983,113297108937174512275

%N Coefficients in the power series A(x) such that: A(x)^3 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

%C Compare to A357153.

%C Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).

%C Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

%H Paul D. Hanna, <a href="/A357163/b357163.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.

%F (1) A(x)^3 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

%F (2) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).

%F (3) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.

%F (4) -A(x)^6 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.

%F (5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*A(x))^(n+1) / A(x)^n.

%F (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+2))^n.

%e G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 313*x^4 + 2834*x^5 + 27088*x^6 + 269380*x^7 + 2757797*x^8 + 28872568*x^9 + 307696566*x^10 + ...

%e such that

%e A(x)^3 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...

%e also

%e -A(x)^6 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...

%o (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);

%o A[#A] = polcoeff(Ser(A)^3 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#A-2); );A[n+1]}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A357153, A357160, A357161, A357162, A357164, A357165.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 17 2022