cp's OEIS Frontend

A361047 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion(x - x^3*A'(x)^2).

%I A361047 #16 Mar 05 2023 07:02:51
%S A361047 1,1,9,159,4051,131688,5132793,231332589,11778989157,666865748751,
%T A361047 41494745678544,2812781975630049,206264308294757115,
%U A361047 16268935714201604701,1373512281722006688063,123601628009085259269819,11812339040349301277253801,1194940136210629914238593762
%N A361047 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion(x - x^3*A'(x)^2).
%C A361047 Conjecture: a(n) == 1 (mod 3) iff n = (3^k - 1)/2 for k >= 0, otherwise a(n) == 0 (mod 3).
%H A361047 Paul D. Hanna, <a href="/A361047/b361047.txt">Table of n, a(n) for n = 1..301</a>
%F A361047 G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
%F A361047 (1) A(x) = Series_Reversion(x - x^3*A'(x)^2).
%F A361047 (2) A(x) = x + A(x)^3 * A'(A(x))^2.
%F A361047 (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(2*n) / n! ).
%F A361047 (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(2*n) / n!, where A'(x) is the g.f. of A361046.
%F A361047 a(n) = A361046(n-1)/(2*n-1) for n >= 1.
%e A361047 G.f.: A(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + ... + a(n)*x^(2*n-1) + ...
%e A361047 By definition, A(x - x^3*A'(x)^2) = x, where
%e A361047 A'(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + A361046(n)*x^(2*n) + ...
%e A361047 Also,
%e A361047 A'(x) = 1 + (d/dx x^3*A'(x)^2) + (d^2/dx^2 x^6*A'(x)^4)/2! + (d^3/dx^3 x^9*A'(x)^6)/3! + (d^4/dx^4 x^12*A'(x)^8)/4! + (d^5/dx^5 x^15*A'(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(2*n))/n! + ...
%e A361047 Further,
%e A361047 A(x) = x * exp( x^2*A'(x)^2 + (d/dx x^5*A'(x)^4)/2! + (d^2/dx^2 x^8*A'(x)^6)/3! + (d^3/dx^3 x^11*A'(x)^8)/4! + (d^4/dx^4 x^14*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(2*n))/n! + ... ).
%o A361047 (PARI) {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*(A')^2 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
%o A361047 for(n=1, 25, print1(a(n), ", "))
%Y A361047 Cf. A361046, A229619, A360977, A360978.
%K A361047 nonn
%O A361047 1,3
%A A361047 _Paul D. Hanna_, Mar 03 2023