A366226 O.g.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} 2^n * log( A(3^n*x) )^n / n!.

1, 1, 6, 261, 56070, 56526498, 334429044030, 15777272891508021, 6500948711591606135796, 22416650201723925643982814186, 632905244163070372226486183732882316, 146120187946706698644410320973489902454862324, 277121097159744219425840626808464318501357604841881466

Offset: 0

Author

Paul D. Hanna, Oct 16 2023

Keywords

• nonn

Comments

In general, we have the following identity:

given A(x) = Sum_{n>=0} a(n)*x^n satisfies

A(x) = 1 + x*Sum_{n>=0} p^n * log( A(q^n*x) )^n / n!,

then a(n+1) = [x^n] A(x)^(p*q^n) for n >= 0, with a(0)=1,

for arbitrary fixed parameters p and q.

Here, p = 2 and q = 3.

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 261*x^3 + 56070*x^4 + 56526498*x^5 + 334429044030*x^6 + 15777272891508021*x^7 + 6500948711591606135796*x^8 + ...
where
A(x) = 1 + x*[1 + 2*log(A(3*x)) + 2^2*log(A(3^2*x))^2/2! + 2^3*log(A(3^2*x))^3/3! + ... + 2^n*log(A(3^n*x))^n/n! + ...].
RELATED SERIES.
log(A(x)) = x + 11*x^2/2 + 766*x^3/3 + 223187*x^4/4 + 282345766*x^5/5 + 2006233236098*x^6/6 + 110438567161208518*x^7/7 + ...
RELATED TABLE.
The table of coefficients of x^k in A(x)^(2*3^n) begins:
n=0: [1,   2,     13,      534,     112698,    113168268, ...];
n=1: [1,   6,     51,     1766,     345165,    340906254, ...];
n=2: [1,  18,    261,     7350,    1112382,   1035922644, ...];
n=3: [1,  54,   1755,    56070,    4589001,   3250238022, ...];
n=4: [1, 162,  14013,   894294,   56526498,  12817431900, ...];
n=5: [1, 486, 120771, 20555046, 2731197285, 334429044030, ...]; ...
in which the main diagonal equals this sequence shift left,
illustrating that a(n+1) = [x^n] A(x)^(2*3^n) for n >= 0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A132695, A156904, A156905, A181444.

Programs

PARI

{a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(2*3^(#A-1)))[ #A])); A[n+1]}
for(n=0,15,print1(a(n),", "))

PARI

{a(n) = my(A=1+x); for(i=1, n, A = 1 + x*sum(m=0,#A, 2^m*log( subst(Ser(A),x,3^m*x +x*O(x^n)))^m/m!) ); polcoeff(A,n)}
for(n=0,15,print1(a(n),", "))

Formula

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

(1) A(x) = 1 + x*Sum_{n>=0} 2^n*log( A(3^n*x) )^n / n!.

(2) a(n+1) = [x^n] A(x)^(2*3^n) for n >= 0, with a(0)=1.