A393888 G.f. A(x) satisfies 3*A(x) = 1 + x + 2*A(x*A(x)).

1, 1, 2, 12, 116, 1488, 23320, 427168, 8903096, 207404528, 5332584624, 149891066560, 4571808534832, 150391362350944, 5308182730319872, 200137985502837504, 8029381213697817040, 341585584249232541760, 15361457031184673748480, 728218200757741770027648, 36297744440787064648161024

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

(1) 3*A(x) = 1 + x + 2*A(x*A(x)).

(2) A(x) = 1 + Sum_{n>=0} 2^n * B^n(x) / 3^(n+1), where B(x) = x*A(x) and B^n(x) denotes the n-th iteration of B(x) with B^0(x) = x.

(3) 2*A(x) = 3*F(x) - x/F(x) - 1, where F(x) = A(x/F(x)) = x/Series_Reversion(x*A(x)).

a(n) ~ c * n^(n + 1/6 + log(3/2)) / (exp(n) * log(3/2)^n), where c = 0.48508736088420566514... - Vaclav Kotesovec, Apr 06 2026

From Seiichi Manyama, Apr 06 2026: (Start)

Let b(n,k) = [x^n] A(x)^k.

b(0,1) = b(1,1) = 1; b(n,1) = 2 * Sum_{j=1..n-1} b(j,1) * b(n-j,j).

For k > 1, b(0,k) = 1; b(n,k) = (1/n) * Sum_{j=1..n} ((k+1)*j-n) * b(j,1) * b(n-j,k).

a(n) = b(n,1). (End)

Example

G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 116*x^4 + 1488*x^5 + 23320*x^6 + 427168*x^7 + 8903096*x^8 + 207404528*x^9 + 5332584624*x^10 + ...
RELATED SERIES.
A(x*A(x)) = 1 + x + 3*x^2 + 18*x^3 + 174*x^4 + 2232*x^5 + 34980*x^6 + 640752*x^7 + 13354644*x^8 + 311106792*x^9 + 7998876936*x^10 + ...
where A(x) = (1 + x)/3 + (2/3)*A(x*A(x)).
If F(x) = A(x/F(x)), then
F(x) = 1 + x + x^2 + 8*x^3 + 75*x^4 + 972*x^5 + 15270*x^6 + 280328*x^7 + 5852107*x^8 + 136506148*x^9 + ...
where 2*A(x) = 3*F(x) - x/F(x) - 1.
RELATED TABLE.
The table of coefficients in the n-th iteration of x*A(x) begins
  n = 0: [1, 0,  0,   0,    0,      0,       0, ...];
  n = 1: [1, 1,  2,  12,  116,   1488,   23320, ...];
  n = 2: [1, 2,  6,  35,  326,   4082,   62932, ...];
  n = 3: [1, 3, 12,  75,  692,   8504,  129064, ...];
  n = 4: [1, 4, 20, 138, 1300,  15870,  237972, ...];
  n = 5: [1, 5, 30, 230, 2260,  27810,  414784, ...];
  n = 6: [1, 6, 42, 357, 3706,  46588,  697064, ...];
  n = 7: [1, 7, 56, 525, 5796,  75222, 1139096, ...];
  ...
in which the following sum along column k equals a(k)
a(1) = 1 = 1/3 + 1*2/3^2 + 1*2^2/3^3 + 1*2^3/3^4 + 1*2^4/3^5 + ...
a(2) = 2 = 0/3 + 1*2/3^2 + 2*2^2/3^3 + 3*2^3/3^4 + 4*2^4/3^5 + ...
a(3) = 12 = 0/3 + 2*2/3^2 + 6*2^2/3^3 + 12*2^3/3^4 + 20*2^4/3^5 + ...
a(4) = 116 = 0/3 + 12*2/3^2 + 35*2^2/3^3 + 75*2^3/3^4 + 138*2^4/3^5 + ...
a(5) = 1488 = 0/3 + 116*2/3^2 + 326*2^2/3^3 + 692*2^3/3^4 + 1300*2^4/3^5 + ...
a(6) = 23320 = 0/3 + 1488*2/3^2 + 4082*2^2/3^3 + 8504*2^3/3^4 + 27810*2^5/3^6 + ...
a(7) = 427168 = 0/3 + 23320*2/3^2 + 62932*2^2/3^3 + 129064*2^3/3^4 + 237972*2^4/3^5 + ...
...

Prog

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1+x + 2*subst(A, x, x*A +x*O(x^n)) - 2*A); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(30)

Crossrefs

Column k=1 of A392379.

Cf. A242003, A393889, A393890.

Sequence in context: A258175 A365575 A225797 * A302286 A035051 A214222

Adjacent sequences: A393885 A393886 A393887 * A393889 A393890 A393891

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2026

Status

approved