A341957 E.g.f. A(x) satisfies: Sum_{n>=0} A(x)^n * exp(2^n*A(x)) / n! = Sum_{n>=0} x^n/(1 - 2^n*x).

1, 1, 8, 144, 4554, 230940, 18177900, 2196712980, 406693854000, 115319921466960, 50017977456121080, 33099984846144881280, 33309128229289401091680, 50790831819884758635873840, 116936359809482874588941613600, 405126119455062475269210516705600

Offset: 1

Author

Paul D. Hanna, Mar 09 2021

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + x^2/2! + 8*x^3/3! + 144*x^4/4! + 4554*x^5/5! + 230940*x^6/6! + 18177900*x^7/7! + 2196712980*x^8/8! + 406693854000*x^9/9! + 115319921466960*x^10/10! + ...
such that
Sum_{n>=0} A(x)^n * exp(2^n*A(x)) / n! = exp(A(x)) + A(x)*exp(2*A(x)) + A(x)^2*exp(2^2*A(x))/2! + A(x)^3*exp(2^3*A(x))/3! + A(x)^4*exp(2^4*A(x))/4! +...
equals the sum
Sum_{n>=0} x^n/(1 - 2^n*x) = 1 + 2*x + 4*x^2 + 10*x^3 + 34*x^4 + 162*x^5 + 1090*x^6 + 10370*x^7 + 139522*x^8 + ... + A117402(n)*x^n + ...
RELATED SERIES.
exp(A(x)) = 1 + x + 2*x^2/2! + 12*x^3/3! + 186*x^4/4! + 5460*x^5/5! + 263940*x^6/6! + 20053740*x^7/7! + 2359326480*x^8/8! + 428122913400*x^9/9! + ...
		

Crossrefs

Cf. A117402.

Programs

PARI

{a(n) = my(L=[0,1]); for(i=1,n, L=concat(L,0);
L[#L] = polcoeff( sum(n=0,#L, x^n/(1 - 2^n*x +x*O(x^#L))) - sum(n=0,#L, Ser(L)^n/n! * exp(2^n*Ser(L)) ) ,#L-1)/2;); n!*L[n+1]}
for(n=1,20,print1(a(n),", "))