A357162 Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

1, 1, 4, 25, 162, 1160, 8731, 68364, 550707, 4535402, 38012170, 323168946, 2780229079, 24158457026, 211721412339, 1869239684558, 16609750957942, 148431230687412, 1333134683364035, 12027524448579488, 108951760865234373, 990555733683233240, 9035754580314840475

Offset: 0

Author

Paul D. Hanna, Sep 17 2022

Keywords

• nonn

Comments

Compare to A357152.

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 + 4*x^2 + 25*x^3 + 162*x^4 + 1160*x^5 + 8731*x^6 + 68364*x^7 + 550707*x^8 + 4535402*x^9 + 38012170*x^10 + ...
such that
A(x)^2 = ... + 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)^5 = ... + 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. A357152, A357160, A357161, A357163, A357164, A357165.

Programs

PARI

{a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A)^2 - 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)^2 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

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

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

(4) -A(x)^5 = 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.