A357161 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

1, 1, 3, 15, 71, 378, 2087, 12006, 70910, 428021, 2627731, 16358961, 103027423, 655236314, 4202210514, 27145925685, 176474644608, 1153679423108, 7579526316199, 50017854059557, 331390828183765, 2203548061830875, 14700363755114949, 98363233394747546

Offset: 0

Author

Paul D. Hanna, Sep 17 2022

Keywords

• nonn

Comments

Compare to A357151.

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

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.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 71*x^4 + 378*x^5 + 2087*x^6 + 12006*x^7 + 70910*x^8 + 428021*x^9 + 2627731*x^10 + ...
such that
A(x) = ... + 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 + ...
also
-A(x)^4 = ... + 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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A357151, A357160, A357162, A357163, A357164, A357165.

Programs

PARI

{a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A) - 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]}
for(n=0,30, print1(a(n),", "))

Formula

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

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

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

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

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

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

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