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A361308 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)).

%I A361308 #12 Mar 25 2025 08:56:10
%S A361308 1,1,8,122,2676,75197,2548336,100461956,4500071172,225305924896,
%T A361308 12456434569184,753380353835754,49473301917640864,3505613955205438686,
%U A361308 266627715169575108168,21667902182055638829520,1873978995774161192935320,171874439346918445003163152
%N A361308 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)).
%H A361308 Paul D. Hanna, <a href="/A361308/b361308.txt">Table of n, a(n) for n = 1..300</a>
%F A361308 G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.
%F A361308 (1) A(x) = Series_Reversion(x - x^4*A'(x)).
%F A361308 (2) A(x) = x + A(x)^4 * A'(A(x)).
%F A361308 (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^n / n! ).
%F A361308 (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^n / n! is the g.f. of A361541.
%F A361308 (5) a(n) = A361541(n-1)/(3*n-2) for n >= 1.
%e A361308 G.f.: A(x) = x + x^4 + 8*x^7 + 122*x^10 + 2676*x^13 + 75197*x^16 + 2548336*x^19 + 100461956*x^22 + 4500071172*x^25 + ... + a(n)*x^(3*n-2) + ...
%e A361308 By definition, A(x - x^4*A'(x)) = x, where
%e A361308 A'(x) = 1 + 4*x^3 + 56*x^6 + 1220*x^9 + 34788*x^12 + 1203152*x^15 + 48418384*x^18 + 2210163032*x^21 + ... + A361541(n)*x^(3*n) + ...
%e A361308 Also,
%e A361308 A'(x) = 1 + (d/dx x^4*A'(x)) + (d^2/dx^2 x^8*A'(x)^2)/2! + (d^3/dx^3 x^12*A'(x)^3)/3! + (d^4/dx^4 x^16*A'(x)^4)/4! + (d^5/dx^5 x^20*A'(x)^5)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^n)/n! + ...
%e A361308 Further,
%e A361308 A(x) = x * exp( x^3*A'(x) + (d/dx x^7*A'(x)^2)/2! + (d^2/dx^2 x^11*A'(x)^3)/3! + (d^3/dx^3 x^15*A'(x)^4)/4! + (d^4/dx^4 x^19*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^n)/n! + ... ).
%o A361308 (PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A' +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
%o A361308 for(n=1, 25, print1(a(n), ", "))
%Y A361308 Cf. A361541.
%Y A361308 Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361309, A361310, A361311.
%K A361308 nonn
%O A361308 1,3
%A A361308 _Paul D. Hanna_, Mar 17 2023