A249475 - OEIS

%I #7 Oct 29 2014 22:19:37

%S 1,1,5,25,156,1048,8400,72384,710184,7519240,87797880,1098513880,

%T 14945280640,216079283040,3352657547680,55071779464352,

%U 961293645943680,17669716422651776,342988501737128576,6978772157389361280,149123855108936024576,3328674238745847019520

%N E.g.f.: exp(2)*P(x) - 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="/A249475/b249475.txt">Table of n, a(n) for n = 0..100</a>

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 156*x^4/4! + 1048*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) = 6.3890560989306502272...

%e q(1) = 6.3890560989306502272...

%e q(2) = 17.167168296791950681...

%e q(3) = 56.279617088237152499...

%e q(4) = 257.78714154011641272...

%e q(5) = 1346.0541760535306736...

%e q(6) = 8772.1663739148311280...

%e q(7) = 63072.176405596679965...

%e q(8) = 527808.01503923686167...

%e q(9) = 4851990.6204200261720...

%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) = 5;

%e a(3) = exp(2)*11 - q(3) = 25;

%e a(4) = exp(2)*56 - q(4) = 156;

%e a(5) = exp(2)*324 - q(5) = 1048;

%e a(6) = exp(2)*2324 - q(6) = 8400;

%e a(7) = exp(2)*18332 - q(7) = 72384;

%e a(8) = exp(2)*167544 - q(8) = 710184; ...

%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, A249476, A249477, A249478, A249480.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 29 2014