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FORMULA
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E.g.f. A(x) satisfies:
(1) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k)*x + (k+1)*A(x) = 0.
(2) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k - m)*x + (k+1)*A(x) = -m/x * A(x)^(2*m-2) / (x + A(x))^(m-2).
(3) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k)*x + (k+1 - p)*A(x) = -p/A(x) * x^(2*p-2) / (x + A(x))^(p-2).
(4) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k - m)*x + (k+1 - p)*A(x) = -(m*x + p*A(x)) * A(x)^(2*m-2) * x^(2*p-2) / (x + A(x))^(m+p-2).
(5) A(A(x)) = x.
(6) Sum_{n>=0} 1/n! * Product_{k=1..n} (n-k)*x + k*A(x) = -A(x)/x.
(7) Sum_{n>=0} 1/n! * Product_{k=1..n} (n-k+1)*x + (k-1)*A(x) = -x/A(x).
(8) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1)*x + (2*k-1)*A(x) = 1.
(9) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1 - m)*x + (2*k-1)*A(x) = (-A(x)/x)^( m*x/(x - A(x)) ).
(10) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1)*x + (2*k-1 - p)*A(x) = (-A(x)/x)^( p*A(x)/(x - A(x)) ).
(11) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1 - m)*x + (2*k-1 - p)*A(x) = (-A(x)/x)^( (m*x + p*A(x))/(x - A(x)) ).
a(n)/n! ~ (-1)^n * c * d^n / n^(3/2), where d = 2.45598128882155545489... and c = 0.2658048623687886... - Vaclav Kotesovec, Jul 12 2018
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EXAMPLE
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E.g.f.: A(x) = -x + 2*x^2/2! - 6*x^3/3! + 36*x^4/4! - 300*x^5/5! + 3270*x^6/6! - 43680*x^7/7! + 691992*x^8/8! - 12670560*x^9/9! + 263281050*x^10/10! - 6119720640*x^11/11! + 157325242140*x^12/12! - 4431909081600*x^13/13! + 135757694361198*x^14/14! - 4492575720622080*x^15/15! + 159723265791222000*x^16/16! + ...
such that
0 = (x + A(x)) + (2*x + A(x))*(x + 2*A(x))/1! + (3*x + A(x))*(2*x + 2*A(x))*(x + 3*A(x))/2! + (4*x + A(x))*(3*x + 2*A(x))*(2*x + 3*A(x))*(x + 4*A(x))/3! + (5*x + A(x))*(4*x + 2*A(x))*(3*x + 3*A(x))*(2*x + 4*A(x))*(x + 5*A(x))/4! + ...
Also,
1 = 1 + (x + A(x))/(1!*2) + (3*x + A(x))*(x + 3*A(x))/(2!*2^2) + (5*x + A(x))*(3*x + 3*A(x))*(x + 5*A(x))/(3!*2^3) + (7*x + A(x))*(5*x + 3*A(x))*(3*x + 5*A(x))*(x + 7*A(x))/(4!*2^4) + ...
EXAMPLES OF SUMS.
The e.g.f. A(x) satisfies the following sums.
(E1) Define
S1(m,p) = Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k - m)*x + (k+1 - p)*A(x),
then
S1(m,p) = -(m*x + p*A(x)) * A(x)^(2*m-2) * x^(2*p-2) / (x + A(x))^(m+p-2).
(E2) Define
S2(m,p) = Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1 - m)*x + (2*k-1 - p)*A(x),
then
S2(m,p) = (-A(x)/x)^( (m*x + p*A(x))/(x - A(x)) ).
Examples of S2(m,p):
S2(0,0) = 1,
S2(1,-1) = -A(x)/x,
S2(-1,1) = -x/A(x),
S2(1,2) = (-A(x)/x)^( (x + 2*A(x))/(x - A(x)) ).
Examples of S1(m,p):
S1(0,0) = 0,
S1(1,0) = -(x + A(x)) / x,
S1(2,0) = -2*A(x)^2 / x,
S1(3,0) = -3*A(x)^4 / (x*(x + A(x))),
S1(0,1) = -(x + A(x)) / A(x),
S1(0,2) = -2*x^2 / A(x),
S1(0,3) = -3*x^4 / (A(x)*(x + A(x))),
S1(1,1) = -(x + A(x)),
S1(2,2) = -2*x^2*A(x)^2 / (x + A(x)),
S1(3,3) = -3*x^4*A(x)^4 / (x + A(x))^3,
S1(2,1) = -A(x)^2 * (2*x + A(x)) / (x + A(x)),
S1(1,2) = -x^2 * (x + 2*A(x)) / (x + A(x)),
S1(4,3) = -x^4*A(x)^6 * (4*x + 3*A(x)) / (x + A(x))^5.
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PROG
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(PARI) {a(n) = my(A=[-1]); for(i=0, n, A=concat(A, 0); A[#A] = -polcoeff( sum(m=0, #A, 1/m!*prod(k=0, m, (m+1-k)*x + (k+1)*x*Ser(A) ) ), #A)); n!*A[n]}
for(n=1, 30, print1(a(n), ", "))
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