A357202 - OEIS

%I #6 Sep 18 2022 12:37:09

%S 1,1,2,9,35,182,921,5062,28234,162330,947773,5622641,33747694,

%T 204676547,1252083028,7717376754,47878314072,298749048454,

%U 1873637869199,11804288518884,74673607921030,474128308291896,3020493580980524,19301224674496592,123681469340775568

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

%C Compare to A357152 and A357162.

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (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="/A357202/b357202.txt">Table of n, a(n) for n = 0..300</a>

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

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

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

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

%F (4) -A(x)^5 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(n+1))^(n+1) / A(x)^n.

%F (5) 0 = Sum_{n=-oo..+oo} x^n * (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)^n.

%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 35*x^4 + 182*x^5 + 921*x^6 + 5062*x^7 + 28234*x^8 + 162330*x^9 + 947773*x^10 + 5622641*x^11 + 33747694*x^12 + ...

%e such that

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

%e also

%e -A(x)^5 = ... + x^(-2)*(A(x) - 1/x)^(-1)*A(x)^2 + x^(-1)*A(x) + (A(x) - x) + x*(A(x) - x^2)^2/A(x) + x^2*(A(x) - x^3)^3/A(x)^2 + x^3*(A(x) - x^4)^4/A(x)^3 + ... + x^n*(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)^2 - sum(n=-#A-2, #A+2, x^(n) * (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. A357152, A357162, A357200, A357201, A357203, A357204, A357205.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 17 2022