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A361310 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^3).

%I A361310 #13 Mar 25 2025 08:56:02
%S A361310 1,1,16,538,26676,1705373,131524408,11778395196,1195433981028,
%T A361310 135247561603456,16853285080609312,2292048750536003426,
%U A361310 337754031605269049112,53608164572529006153454,9118712400086550140230888,1655104918901340697851158384,319341008921919836189242604080
%N A361310 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^3).
%H A361310 Paul D. Hanna, <a href="/A361310/b361310.txt">Table of n, a(n) for n = 1..200</a>
%F A361310 G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.
%F A361310 (1) A(x) = Series_Reversion(x - x^4*A'(x)^3).
%F A361310 (2) A(x) = x + A(x)^4 * A'(A(x))^3.
%F A361310 (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^(3*n) / n! ).
%F A361310 (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^(3*n) / n! is the g.f. of A361543.
%F A361310 (5) a(n) = A361543(n-1)/(3*n-2) for n >= 1.
%e A361310 G.f.: A(x) = x + x^4 + 16*x^7 + 538*x^10 + 26676*x^13 + 1705373*x^16 + 131524408*x^19 + 11778395196*x^22 + ... + a(n)*x^(3*n-2) + ...
%e A361310 By definition, A(x - x^4*A'(x)^3) = x, where
%e A361310 A'(x) = 1 + 4*x^3 + 112*x^6 + 5380*x^9 + 346788*x^12 + 27285968*x^15 + 2498963752*x^18 + 259124694312*x^21 + ... + A361543(n)*x^(3*n) + ...
%e A361310 Also,
%e A361310 A'(x) = 1 + (d/dx x^4*A'(x)^3) + (d^2/dx^2 x^8*A'(x)^6)/2! + (d^3/dx^3 x^12*A'(x)^9)/3! + (d^4/dx^4 x^16*A'(x)^12)/4! + (d^5/dx^5 x^20*A'(x)^15)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^(3*n))/n! + ...
%e A361310 Further,
%e A361310 A(x) = x * exp( x^3*A'(x)^3 + (d/dx x^7*A'(x)^6)/2! + (d^2/dx^2 x^11*A'(x)^9)/3! + (d^3/dx^3 x^15*A'(x)^12)/4! + (d^4/dx^4 x^19*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^(3*n))/n! + ... ).
%o A361310 (PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A'^3 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
%o A361310 for(n=1, 25, print1(a(n), ", "))
%Y A361310 Cf. A361543.
%Y A361310 Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361309, A361311.
%K A361310 nonn
%O A361310 1,3
%A A361310 _Paul D. Hanna_, Mar 17 2023