A393889 G.f. A(x) satisfies 4*A(x) = 1 + x + 3*A(x*A(x)).

1, 1, 3, 27, 387, 7317, 168237, 4503789, 136727055, 4625789067, 172276066539, 6998043525885, 307831417695909, 14577640870303377, 739536100222798899, 40020776276242083507, 2301691209030102539259, 140217912528862509036117, 9021108445527923796514437, 611285061740709140815872789

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

(2) A(x) = 1 + Sum_{n>=0} 3^n * B^n(x) / 4^(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) 3*A(x) = 4*F(x) - x/F(x) - 1, where F(x) = A(x/F(x)) = x/Series_Reversion(x*A(x)).

a(n) ~ c * n^(n + 1/4 + 2*log(4/3)) / (exp(n) * log(4/3)^n), where c = 0.28592046327618102388... - 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) = 3 * 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 + 3*x^2 + 27*x^3 + 387*x^4 + 7317*x^5 + 168237*x^6 + 4503789*x^7 + 136727055*x^8 + 4625789067*x^9 + ...
RELATED SERIES.
A(x*A(x)) = 1 + x + 4*x^2 + 36*x^3 + 516*x^4 + 9756*x^5 + 224316*x^6 + 6005052*x^7 + 182302740*x^8 + 6167718756*x^9 + ...
where A(x) = (1 + x)/4 + (3/4)*A(x*A(x)).
If F(x) = A(x/F(x)), then
F(x) = 1 + x + 2*x^2 + 20*x^3 + 286*x^4 + 5426*x^5 + 124968*x^6 + 3349446*x^7 + 101773794*x^8 + 3445660160*x^9 + ...
where 3*A(x) = 4*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,  3,  27,   387,   7317,  168237, ...];
  n = 2: [1, 2,  8,  70,   978,  18192,  413694, ...];
  n = 3: [1, 3, 15, 135,  1861,  34133,  767905, ...];
  n = 4: [1, 4, 24, 228,  3148,  57196, 1274540, ...];
  n = 5: [1, 5, 35, 355,  4975,  90105, 1993293, ...];
  n = 6: [1, 6, 48, 522,  7502, 136372, 3004490, ...];
  n = 7: [1, 7, 63, 735, 10913, 200417, 4414417, ...];
  ...
in which the following sum along column k equals a(k)
a(1) = 1 = 1/4 + 1*3^1/4^2 + 1*3^2/4^3 + 1*3^3/4^4 + 1*3^4/4^5 + ...
a(2) = 3 = 0/4 + 1*3^1/4^2 + 2*3^2/4^3 + 3*3^3/4^4 + 4*3^4/4^5 + ...
a(3) = 27 = 0/4 + 3*3^1/4^2 + 8*3^2/4^3 + 15*3^3/4^4 + 24*3^4/4^5 + ...
a(4) = 387 = 0/4 + 27*3^1/4^2 + 70*3^2/4^3 + 135*3^3/4^4 + 228*3^4/4^5 + ...
a(5) = 7317 = 0/4 + 387*3^1/4^2 + 978*3^2/4^3 + 1861*3^3/4^4 + 3148*3^4/4^5 + ...
a(6) = 168237 = 0/4 + 7317*3^1/4^2 + 18192*3^2/4^3 + 34133*3^3/4^4 + 57196*3^4/4^5 + ...
a(7) = 4503789 = 0/4 + 168237*3^1/4^2 + 413694*3^2/4^3 + 767905*3^3/4^4 + 1274540*3^4/4^5 + ...
...

Prog

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1+x + 3*subst(A, x, x*A +x*O(x^n)) - 3*A); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(30)

Crossrefs

Column k=1 of A394047.

Cf. A242003, A393888, A393890.

Sequence in context: A365794 A138436 A141057 * A365569 A365586 A201696

Adjacent sequences: A393886 A393887 A393888 * A393890 A393891 A393892

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2026

Status

approved