A393856 G.f. A(x) satisfies: A( x - x*A(4*x)/4 ) = x.

1, 1, 6, 121, 8466, 2227706, 2299117036, 9437900978761, 154724962114074266, 10141757093759238322206, 2658722471751725008735417876, 2787907137799972478301903768489866, 11693369090347081889573726480398459556596, 196182354396929228629160971993147949773294608596, 13165578079340528240926058991817466632628777784210130776

Offset

1,3

Comments

Conjecture: a(n) is odd iff n is a power of 2.

Conjecture: a(n) == 1 (mod 5) for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Formula

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

(1) A( x - x*A(4*x)/4 ) = x.

(2) A(x) = INV(x - x*INV(x - 4*x*INV(x - 4^2*x*INV(x - 4^3*x*INV(x - 4^4*x*INV(x - ...)))))), where INV(F(x)) = series reversion of F(x).

(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * (A(4*x)/4)^n / n!.

(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * (A(4*x)/4)^n / n! ).

Example

G.f.: A(x) = x + x^2 + 6*x^3 + 121*x^4 + 8466*x^5 + 2227706*x^6 + 2299117036*x^7 + 9437900978761*x^8 + ...
The g.f. A(x) results from infinitely nested series reversions as shown below.
Series_Reversion(A(x)) = x - x*B(x) where
B(x) = x + 4*x^2 + 96*x^3 + 7744*x^4 + 2167296*x^5 + 2281170944*x^6 + ...
Series_Reversion(B(x)) = x - 4*x*C(x) where
C(x) = x + 16*x^2 + 1536*x^3 + 495616*x^4 + 554827776*x^5 + ...
Series_Reversion(C(x)) = x - 4^2*x*D(x) where
D(x) = x + 64*x^2 + 24576*x^3 + 31719424*x^4 + 142035910656*x^5 + ...
Series_Reversion(D(x)) = x - 4^3*x*E(x) where
E(x) = x + 256*x^2 + 393216*x^3 + 2030043136*x^4 + 36361193127936*x^5 + ...
...

Prog

(PARI) {a(n, k=4) = my(A=x); for(i=1, n, A = serreverse(x - x*subst(A, x, k*x)/k +x*O(x^n)); ); polcoef(GF=A, n)}
{upto(n, k=4) = a(n, k); Vec(GF)}
upto(20)
(PARI) {a(n, k=4) = my(A=x+x^2); for(j=0, n, A = serreverse(x - k^(n-j)*x*A + x*O(x^n))); polcoef(GF=A, n)}
{upto(n, k=4) = a(n, k); Vec(GF)}
upto(20)

Crossrefs

Cf. A000108 (k=0), A088714 (k=1), A195195 (k=2), A195196 (k=3).

Cf. A393857 (k=5), A393858 (k=6), A393859 (k=7).

Sequence in context: A336389 A126244 A336808 * A393693 A296801 A249168

Adjacent sequences: A393852 A393853 A393854 * A393857 A393858 A393859

Keyword

nonn

Author

Paul D. Hanna, Mar 11 2026

Status

approved