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A361311 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^5*A'(x)).

%I A361311 #11 Mar 25 2025 08:55:57
%S A361311 1,1,10,195,5520,201255,8881551,457227585,26805712005,1759840463070,
%T A361311 127784731466660,10164274303786460,878859905526721250,
%U A361311 82080454974318915935,8235485665033295289810,883569144560890419421630,100952601749463417250801935,12239031817482031919864850550
%N A361311 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^5*A'(x)).
%H A361311 Paul D. Hanna, <a href="/A361311/b361311.txt">Table of n, a(n) for n = 1..200</a>
%F A361311 G.f. A(x) = Sum_{n>=1} a(n)*x^(4*n-3) may be defined by the following.
%F A361311 (1) A(x) = Series_Reversion(x - x^5*A'(x)).
%F A361311 (2) A(x) = x + A(x)^5 * A'(A(x)).
%F A361311 (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(5*n-1) * A'(x)^n / n! ).
%F A361311 (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(5*n) * A'(x)^n / n! is the g.f. of A361551.
%F A361311 (5) a(n) = A361551(n-1)/(4*n-3) for n >= 1.
%e A361311 G.f.: A(x) = x + x^5 + 10*x^9 + 195*x^13 + 5520*x^17 + 201255*x^21 + 8881551*x^25 + 457227585*x^29 + ... + a(n)*x^(4*n-3) + ...
%e A361311 By definition, A(x - x^5*A'(x)) = x, where
%e A361311 A'(x) = 1 + 5*x^4 + 90*x^8 + 2535*x^12 + 93840*x^16 + 4226355*x^20 + 222038775*x^24 + ... + A361551(n)*x^(4*n) + ...
%e A361311 Also,
%e A361311 A'(x) = 1 + (d/dx x^5*A'(x)) + (d^2/dx^2 x^10*A'(x)^2)/2! + (d^3/dx^3 x^15*A'(x)^3)/3! + (d^4/dx^4 x^20*A'(x)^4)/4! + (d^5/dx^5 x^25*A'(x)^5)/5! + ... + (d^n/dx^n x^(5*n)*A'(x)^n)/n! + ...
%e A361311 Further,
%e A361311 A(x) = x * exp( x^4*A'(x) + (d/dx x^9*A'(x)^2)/2! + (d^2/dx^2 x^14*A'(x)^3)/3! + (d^3/dx^3 x^19*A'(x)^4)/4! + (d^4/dx^4 x^24*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(5*n-1)*A'(x)^n)/n! + ... ).
%o A361311 (PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^5*A' +x*O(x^(4*n)))); polcoeff(A, 4*n-3)}
%o A361311 for(n=1, 25, print1(a(n), ", "))
%Y A361311 Cf. A361551.
%Y A361311 Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361309, A361310.
%K A361311 nonn
%O A361311 1,3
%A A361311 _Paul D. Hanna_, Mar 17 2023