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E.g.f.: A(x) = 1 + x + x^2/2! + 5*x^3/3! + 12*x^4/4! + 88*x^5/5! +...
such that A(x) = exp(-2)*P(x) + Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} -(-2)^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 2/(1-x) - 2^2/((1-x)*(2-x^2)) + 2^3/((1-x)*(2-x^2)*(3-x^3)) - 2^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 2^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 0.864664716763387308106000...
q(1) = 0.864664716763387308106000...
q(2) = 0.593994150290161924318001...
q(3) = 3.511311884397260389166005...
q(4) = 4.421224138749689253936028...
q(5) = 44.15136823133748782634416...
q(6) = 181.4808017581121040383451...
q(7) = 1551.033587706416132199201...
q(8) = 9397.385305404963149311748...
q(9) = 108557.0073471358880187848...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(-2)*1 + q(0) = 1;
a(1) = exp(-2)*1 + q(1) = 1;
a(2) = exp(-2)*3 + q(2) = 1;
a(3) = exp(-2)*11 + q(3) = 5;
a(4) = exp(-2)*56 + q(4) = 12;
a(5) = exp(-2)*324 + q(5) = 88;
a(6) = exp(-2)*2324 + q(6) = 496;
a(7) = exp(-2)*18332 + q(7) = 4032;
a(8) = exp(-2)*167544 + q(8) = 32072; ...
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