A380063 G.f. A(x) satisfies 1 + 4*A(x) = Sum_{n>=0} (x + 3*A(x)^n)^n.

1, 1, 7, 22, 136, 682, 4030, 23539, 143026, 883696, 5546230, 35293039, 226922620, 1473059344, 9637454320, 63493041340, 420841313770, 2804372957419, 18777007427668, 126261555174991, 852293214690055, 5773260342048436, 39231225884581288, 267363603092143528, 1826957733209857000, 12514655218275667486

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) 1 + 4*A(x) = Sum_{n>=0} (x + 3*A(x)^n)^n.

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

a(n) ~ c * d^n / n^(3/2), where d = 7.269242474387887958966332897... and c = 0.0653173269002661770398318... - Vaclav Kotesovec, Jan 22 2025

Example

G.f.: A(x) = x + x^2 + 7*x^3 + 22*x^4 + 136*x^5 + 682*x^6 + 4030*x^7 + 23539*x^8 + 143026*x^9 + 883696*x^10 + 5546230*x^11 + 35293039*x^12 + ...
where
1 + 4*A(x) = 1 + (x + 3*A(x)) + (x + 3*A(x)^2)^2 + (x + 3*A(x)^3)^3 + (x + 3*A(x)^4)^4 + (x + 3*A(x)^5)^5 + ...
also
1 + 4*A(x) = 1/(1 - x) + 3*A(x)/(1 - x*A(x))^2 + 3^2*A(x)^4/(1 - x*A(x)^2)^3 + 3^3*A(x)^9/(1 - x*A(x)^3)^4 + 3^4*A(x)^16/(1 - x*A(x)^4)^5 + ...
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.13752764671682708197969294468903369545117753579440...
  where 2 = Sum_{n>=0} (t + 3/4^n)^n.
A(t) = 1/5 at t = 0.13218979829105822065435631622519782747342694822528...
  where 9/5 = Sum_{n>=0} (t + 3/5^n)^n.
A(t) = 1/6 at t = 0.12181389616442597666239358061942842843956426674807...
  where 5/3 = Sum_{n>=0} (t + 3/6^n)^n.
A(t) = 1/7 at t = 0.11121584078034746775788323981995465764427116561334...
A(t) = 1/8 at t = 0.10159440637613508856137826877971867899596426901620...
  where 3/2 = Sum_{n>=0} (t + 3/8^n)^n.
A(1/8) = 0.17527526517934745246924428431105068622657459820748...
  where 1 + 4*A(1/8) = Sum_{n>=0} (1/8 + 3*A(1/8)^n)^n.
A(1/9) = 0.14264756148283566210339311854398565959288119486414...
A(1/10) = 0.12226877852777625938200787071716322394161142457576...

Prog

(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(n=1, #V, (x + 3*A^n)^n ) - 4*A, #V-1) ); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A380061, A380062, A380064.

Sequence in context: A117110 A027838 A194996 * A330089 A000835 A239028

Adjacent sequences: A380060 A380061 A380062 * A380064 A380065 A380066

Keyword

nonn

Author

Paul D. Hanna, Jan 11 2025

Status

approved