A368631 Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)^2).

1, 1, 7, 118, 3457, 150376, 8869249, 669261160, 62084355505, 6878901271024, 890797404903841, 132568595259161656, 22370325575395442473, 4233795107469842535544, 890606081738110684972705, 206651730919408572588445216, 52550877215770005095599441249, 14564273590596678338725804835680

Offset

0,3

Comments

Conjecture: a(n) == 1 (mod 3) for n >= 0.

Conjecture: a(2*n) == 1 (mod 2) for n >= 0.

Conjecture: a(2*n+1) == 0 (mod 2) for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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

(1) A(x/A(x)) = exp(x*A(x)^2).

(2) A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)).

(3) A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)^2) = ( x/Series_Reversion(x*A(x)^2) )^(1/2).

Example

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 118*x^3/3! + 3457*x^4/4! + 150376*x^5/5! + 8869249*x^6/6! + 669261160*x^7/7! + 62084355505*x^8/8! + ...
where A(x/A(x)) = exp(x*A(x)^2) and
exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 61*x^3/3! + 1377*x^4/4! + 49001*x^5/5! + 2476273*x^6/6! + 165555909*x^7/7! + ...
Also,
A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)) begins
B(x) = 1 + x + 9*x^2/2! + 187*x^3/3! + 6461*x^4/4! + 320721*x^5/5! + 21079255*x^6/6! + 1741882717*x^7/7! + ...
B(x)^3 = 1 + 3*x + 33*x^2/2! + 729*x^3/3! + 25653*x^4/4! + 1275483*x^5/5! + 83368251*x^6/6! + ...
Further,
A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)^2) begins
C(x) = 1 + x + 3*x^2/2! + 34*x^3/3! + 809*x^4/4! + 30336*x^5/5! + 1584517*x^6/6! + 107443540*x^7/7! + ...
C(x)^2 = 1 + 2*x + 8*x^2/2! + 86*x^3/3! + 1944*x^4/4! + 70802*x^5/5! + 3628996*x^6/6! + ...

Prog

(PARI) {a(n) = my(A=1+x); for(i=0, n, A = exp( x*((1/x)*serreverse( x/(A + x*O(x^n)) ))^3 )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A368630, A368632, A144682, A144683, A367385.

Sequence in context: A213112 A376035 A266482 * A076283 A057769 A221031

Adjacent sequences: A368628 A368629 A368630 * A368632 A368633 A368634

Keyword

nonn

Author

Paul D. Hanna, Jan 02 2024

Status

approved