A393856 - OEIS

%I #14 Mar 12 2026 10:39:38

%S 1,1,6,121,8466,2227706,2299117036,9437900978761,154724962114074266,

%T 10141757093759238322206,2658722471751725008735417876,

%U 2787907137799972478301903768489866,11693369090347081889573726480398459556596,196182354396929228629160971993147949773294608596,13165578079340528240926058991817466632628777784210130776

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

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

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

%H Paul D. Hanna, <a href="/A393856/b393856.txt">Table of n, a(n) for n = 1..100</a>

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

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

%F (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).

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

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

%e 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 + ...

%e The g.f. A(x) results from infinitely nested series reversions as shown below.

%e Series_Reversion(A(x)) = x - x*B(x) where

%e B(x) = x + 4*x^2 + 96*x^3 + 7744*x^4 + 2167296*x^5 + 2281170944*x^6 + ...

%e Series_Reversion(B(x)) = x - 4*x*C(x) where

%e C(x) = x + 16*x^2 + 1536*x^3 + 495616*x^4 + 554827776*x^5 + ...

%e Series_Reversion(C(x)) = x - 4^2*x*D(x) where

%e D(x) = x + 64*x^2 + 24576*x^3 + 31719424*x^4 + 142035910656*x^5 + ...

%e Series_Reversion(D(x)) = x - 4^3*x*E(x) where

%e E(x) = x + 256*x^2 + 393216*x^3 + 2030043136*x^4 + 36361193127936*x^5 + ...

%e ...

%o (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)}

%o {upto(n,k=4) = a(n,k); Vec(GF)}

%o upto(20)

%o (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)}

%o {upto(n,k=4) = a(n,k); Vec(GF)}

%o upto(20)

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

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

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 11 2026