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%I #18 Oct 29 2014 22:13:49
%S 1,1,4,17,96,595,4516,37104,351020,3604001,41007240,502039444,
%T 6703536516,95376507135,1459072099824,23677731306350,408821193129564,
%U 7443839953433701,143258713990271960,2893053522512463984,61396438056305204020,1362146168353191078195,31605702195327725326560
%N E.g.f.: exp(1)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 1/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="/A249078/b249078.txt">Table of n, a(n) for n = 0..100</a>
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 96*x^4/4! + 595*x^5/5! +...
%e such that A(x) = exp(1)*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} 1/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) = 1/(1-x) + 1/((1-x)*(2-x^2)) + 1/((1-x)*(2-x^2)*(3-x^3)) + 1/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 1/((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) = 1.7182818284590452...
%e q(1) = 1.7182818284590452...
%e q(2) = 4.1548454853771357...
%e q(3) = 12.901100113049497...
%e q(4) = 56.223782393706533...
%e q(5) = 285.72331242073065...
%e q(6) = 1801.2869693388211...
%e q(7) = 12727.542479311217...
%e q(8) = 104411.81066734227...
%e q(9) = 947120.40724315491...
%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(1)*1 - q(0) = 1;
%e a(1) = exp(1)*1 - q(1) = 1;
%e a(2) = exp(1)*3 - q(2) = 4;
%e a(3) = exp(1)*11 - q(3) = 17;
%e a(4) = exp(1)*56 - q(4) = 96;
%e a(5) = exp(1)*324 - q(5) = 595;
%e a(6) = exp(1)*2324 - q(6) = 4516;
%e a(7) = exp(1)*18332 - q(7) = 37104;
%e a(8) = exp(1)*167544 - q(8) = 351020; ...
%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,prod(k=1,n,1./(k-x^k +O(x^31))))));}
%o for(n=0,30,print1(round(exp(1)*P[n+1]-Q[n+1]),", "))
%Y Cf. A007841, A249474, A249475, A249476, A249477, A249478, A249480.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 28 2014
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