A249475 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).

1, 1, 5, 25, 156, 1048, 8400, 72384, 710184, 7519240, 87797880, 1098513880, 14945280640, 216079283040, 3352657547680, 55071779464352, 961293645943680, 17669716422651776, 342988501737128576, 6978772157389361280, 149123855108936024576, 3328674238745847019520

Offset: 0

Author

Paul D. Hanna, Oct 29 2014

Keywords

• nonn

Comments

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.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 156*x^4/4! + 1048*x^5/5! +...
such that A(x) = exp(2)*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} 2^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) = 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)) +...
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) = 6.3890560989306502272...
q(1) = 6.3890560989306502272...
q(2) = 17.167168296791950681...
q(3) = 56.279617088237152499...
q(4) = 257.78714154011641272...
q(5) = 1346.0541760535306736...
q(6) = 8772.1663739148311280...
q(7) = 63072.176405596679965...
q(8) = 527808.01503923686167...
q(9) = 4851990.6204200261720...
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(2)*1 - q(0) = 1;
a(1) = exp(2)*1 - q(1) = 1;
a(2) = exp(2)*3 - q(2) = 5;
a(3) = exp(2)*11 - q(3) = 25;
a(4) = exp(2)*56 - q(4) = 156;
a(5) = exp(2)*324 - q(5) = 1048;
a(6) = exp(2)*2324 - q(6) = 8400;
a(7) = exp(2)*18332 - q(7) = 72384;
a(8) = exp(2)*167544 - q(8) = 710184; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A007841, A249078, A249474, A249476, A249477, A249478, A249480.

Programs

PARI

\p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, 2^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(2)*P[n+1]-Q[n+1]), ", "))