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Expansion of g.f. A(x) satisfying A(x) = x * Sum_{n>=0} d^n/dx^n x^(2*n-1) * A(x)^n / n!.
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%I #13 Feb 24 2023 02:30:46

%S 1,1,5,37,353,4061,54221,820205,13829377,256853629,5208050365,

%T 114465346733,2711004465185,68846143222013,1866577974450733,

%U 53824099877628077,1645120108520147713,53135285623703158429,1808560829585046118685,64707781796679229092045,2428043851750587122468513

%N Expansion of g.f. A(x) satisfying A(x) = x * Sum_{n>=0} d^n/dx^n x^(2*n-1) * A(x)^n / n!.

%H Paul D. Hanna, <a href="/A356848/b356848.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows.

%F (1) A(x) = x * Sum_{n>=0} d^n/dx^n x^(2*n-1) * A(x)^n / n!.

%F (2) B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^n / n! ) where B(x - x^2*A(x)) = x, and B(x) is the g.f. of A360579.

%F a(n) ~ c * n! * n^(2*LambertW(1)) / LambertW(1)^n, where c = 0.421381125316... - _Vaclav Kotesovec_, Feb 23 2023

%e G.f.: A(x) = 1 + x + 5*x^2 + 37*x^3 + 353*x^4 + 4061*x^5 + 54221*x^6 + 820205*x^7 + 13829377*x^8 + 256853629*x^9 + 5208050365*x^10 + ...

%e such that

%e A(x) = x * (1/x + d/dx x*A(x)) + (d^2/dx^2 x^3*A(x)^2)/2! + (d^3/dx^3 x^5*A(x)^3)/3! + (d^4/dx^4 x^7*A(x)^4)/4! + (d^5/dx^5 x^9*A(x)^5)/5! + (d^6/dx^6 x^11*A(x)^6)/6! + ... + (d^n/dx^n x^(2*n-1)*A(x)^n)/n! + ...).

%e Related series.

%e Let B(x) satisfy B( x - x^2*A(x) ) = x, then

%e B(x) = x * exp( x*A(x) + (d/dx x^3*A(x)^2)/2! + (d^2/dx^2 x^5*B(x)^3)/3! + (d^3/dx^3 x^7*B(x)^4)/4! + (d^4/dx^4 x^9*B(x)^5)/5! + (d^5/dx^5 x^11*B(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n)/n! + ...)

%e where B(x) is the g.f. of A360579:

%e B(x) = x + x^2 + 3*x^3 + 15*x^4 + 105*x^5 + 941*x^6 + 10227*x^7 + 130103*x^8 + 1890785*x^9 + 30848357*x^10 + ... + A360579(n)*x^n + ...

%e and

%e Series_Reversion(B(x)) = x - x^2 - x^3 - 5*x^4 - 37*x^5 - 353*x^6 - 4061*x^7 - 54221*x^8 + ... + -a(n)*x^(n+2) + ...

%o (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n) = my(A=1); for(i=1, n, A = 1 + x*sum(m=1, n, Dx(m, x^(2*m-1)*A^m/m!)) +O(x^(n+4))); polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A360579.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 23 2023