A366227 O.g.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.

1, 1, 6, 138, 8648, 1272948, 424058592, 334836466656, 728593565874816, 5632989888855720864, 184539760855097635059200, 25027477244647424010315231744, 13206715998089387470949589465286656, 26431031766456352400292737393044784872448, 199091399877503863934385670788355318673030504448

Offset: 0

Author

Paul D. Hanna, Oct 17 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 = 3 and q = 2.

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 138*x^3 + 8648*x^4 + 1272948*x^5 + 424058592*x^6 + 334836466656*x^7 + 728593565874816*x^8 + ...
where
A(x) = 1 + x*[1 + 3*log(A(2*x)) + 3^2*log(A(2^2*x))^2/2! + 3^3*log(A(2^2*x))^3/3! + ... + 3^n*log(A(2^n*x))^n/n! + ...].
RELATED SERIES.
log(A(x)) = x + 11*x^2/2 + 397*x^3/3 + 33991*x^4/4 + 6318201*x^5/5 + 2536406543*x^6/6 + 2340834765809*x^7/7 + ...
RELATED TABLE.
The table of coefficients of x^k in A(x)^(3*2^n) begins:
n=0: [1,   3,    21,     451,    26898,    3876222, ...];
n=1: [1,   6,    51,    1028,    56943,    7932774, ...];
n=2: [1,  12,   138,    2668,   128823,   16653720, ...];
n=3: [1,  24,   420,    8648,   340722,   37135560, ...];
n=4: [1,  48,  1416,   37456,  1272948,   97890096, ...];
n=5: [1,  96,  5136,  210848,  8146728,  424058592, ...];
n=6: [1, 192, 19488, 1407808, 83154768, 4578119616, 334836466656, ...]; ...
in which the main diagonal equals this sequence shift left,
illustrating that a(n+1) = [x^n] A(x)^(3*2^n) for n >= 0.
		

Crossrefs

Cf. A132695, A156904, A156905, A181444, A366226.

Programs

PARI

{a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(3*2^(#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, 3^m*log( subst(Ser(A), x, 2^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} 3^n * log( A(2^n*x) )^n / n!.

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