A249474 - OEIS

%I #15 Oct 29 2014 22:14:45

%S 1,1,2,7,30,169,1128,8700,76494,753139,8182188,97131376,1256860330,

%T 17470791933,261284377168,4164406202270,70677340199670,

%U 1268718107324255,24091289738163140,480954355282406340,10097484764045220626,221918808641500960217,5103937368681669463800

%N E.g.f.: P(x)/exp(1) + Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} -(-1)^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="/A249474/b249474.txt">Table of n, a(n) for n = 0..200</a>

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 30*x^4/4! + 169*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)^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) = 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) = 0.632120558828557678...

%e q(1) = 0.632120558828557678...

%e q(2) = 0.896361676485673035...

%e q(3) = 2.953326147114134462...

%e q(4) = 9.398751294399229990...

%e q(5) = 49.80706106045268780...

%e q(6) = 273.0481787175680446...

%e q(7) = 1956.034084445119360...

%e q(8) = 14858.00690837186767...

%e q(9) = 137211.6953065362928...

%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) = 2;

%e a(3) = exp(-1)*11 + q(3) = 7;

%e a(4) = exp(-1)*56 + q(4) = 30;

%e a(5) = exp(-1)*324 + q(5) = 169;

%e a(6) = exp(-1)*2324 + q(6) = 1128;

%e a(7) = exp(-1)*18332 + q(7) = 8700;

%e a(8) = exp(-1)*167544 + q(8) = 76494; ...

%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, -(-1)^n * 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, A249078, A249475, A249476, A249477, A249478, A249480.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 29 2014