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%I #7 Oct 29 2014 22:02:25
%S 1,1,1,5,12,88,496,4032,32072,335144,3443928,41477176,523289472,
%T 7298441952,107525078304,1714360202528,28771306555776,515446334184832,
%U 9722819034952832,193501572577378944,4042243606465206784,88584621284011603968,2029364250844776170496,48539531534286294782976
%N E.g.f.: P(x)/exp(2) + Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} -(-2)^n/Product_{k=1..n} (k - x^k).
%C The function P(x) = Product_{n>=1} 1/(1 - x^n/n) equals the e.g.f. of A007841, the number of factorizations of permutations of n letters into cycles in nondecreasing length order.
%H Paul D. Hanna, <a href="/A249478/b249478.txt">Table of n, a(n) for n = 0..200</a>
%e E.g.f.: A(x) = 1 + x + x^2/2! + 5*x^3/3! + 12*x^4/4! + 88*x^5/5! +...
%e such that A(x) = exp(-2)*P(x) + Q(x), where
%e P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
%e Q(x) = Sum_{n>=1} -(-2)^n / Product_{k=1..n} (k - x^k).
%e More explicitly,
%e P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
%e 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)) +...
%e We can illustrate the initial terms a(n) in the following manner.
%e The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
%e q(0) = 0.864664716763387308106000...
%e q(1) = 0.864664716763387308106000...
%e q(2) = 0.593994150290161924318001...
%e q(3) = 3.511311884397260389166005...
%e q(4) = 4.421224138749689253936028...
%e q(5) = 44.15136823133748782634416...
%e q(6) = 181.4808017581121040383451...
%e q(7) = 1551.033587706416132199201...
%e q(8) = 9397.385305404963149311748...
%e q(9) = 108557.0073471358880187848...
%e and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
%e A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
%e from which we can generate this sequence like so:
%e a(0) = exp(-2)*1 + q(0) = 1;
%e a(1) = exp(-2)*1 + q(1) = 1;
%e a(2) = exp(-2)*3 + q(2) = 1;
%e a(3) = exp(-2)*11 + q(3) = 5;
%e a(4) = exp(-2)*56 + q(4) = 12;
%e a(5) = exp(-2)*324 + q(5) = 88;
%e a(6) = exp(-2)*2324 + q(6) = 496;
%e a(7) = exp(-2)*18332 + q(7) = 4032;
%e a(8) = exp(-2)*167544 + q(8) = 32072; ...
%o (PARI) \p100 \\ set precision
%o {P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
%o {Q=Vec(serlaplace(sum(n=1, 201, -(-2)^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
%o for(n=0, 30, print1(round(exp(-2)*P[n+1]+Q[n+1]), ", "))
%Y Cf. A007841, A249078, A249474, A249475, A249476, A249477, A249480.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Oct 29 2014
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