E.g.f.: A(x) = 1 + x + 6*x^2/2! + 35*x^3/3! + 242*x^4/4! + 1773*x^5/5! +...
such that A(x) = exp(3)*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} 3^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) = 3/(1-x) + 3^2/((1-x)*(2-x^2)) + 3^3/((1-x)*(2-x^2)*(3-x^3)) + 3^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 3^5/((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) = 19.085536923187667740...
q(1) = 19.085536923187667740...
q(2) = 54.256610769563003222...
q(3) = 185.94090615506434515...
q(4) = 882.79006769850939349...
q(5) = 4734.7139631128043480...
q(6) = 31622.787809488139829...
q(7) = 231556.06287587632502...
q(8) = 1971489.1982585546039...
q(9) = 18370572.391163877342...
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(3)*1 - q(0) = 1;
a(1) = exp(3)*1 - q(1) = 1;
a(2) = exp(3)*3 - q(2) = 6;
a(3) = exp(3)*11 - q(3) = 35;
a(4) = exp(3)*56 - q(4) = 242;
a(5) = exp(3)*324 - q(5) = 1773;
a(6) = exp(3)*2324 - q(6) = 15056;
a(7) = exp(3)*18332 - q(7) = 136652;
a(8) = exp(3)*167544 - q(8) = 1393722; ...
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