A359713 a(n) = coefficient of x^n in A(x) such that 3 = Sum_{n=-oo..+oo} (-x)^n * (3*A(x) + x^(n-1))^(n+1).

1, 5, 31, 206, 1433, 10329, 76459, 577855, 4440538, 34591555, 272545144, 2168118299, 17390330046, 140486973983, 1142036572271, 9335129425718, 76681549612006, 632655728172281, 5240339959916895, 43561574812700958, 363294379940353624, 3038799803831856805

Offset: 0

Author

Paul D. Hanna, Jan 17 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 5*x + 31*x^2 + 206*x^3 + 1433*x^4 + 10329*x^5 + 76459*x^6 + 577855*x^7 + 4440538*x^8 + 34591555*x^9 + 272545144*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A359670, A359711, A359712, A363104, A363105.

Programs

PARI

{a(n) = my(A=1,y=3); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A,n,x)}
for(n=0,25, print1( a(n),", "))

PARI

{a(n) = my(A=[1],y=3); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.

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

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

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

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

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

a(n) = Sum_{k=0..n} A359670(n,k)*3^k for n >= 0.