A249474 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).

1, 1, 2, 7, 30, 169, 1128, 8700, 76494, 753139, 8182188, 97131376, 1256860330, 17470791933, 261284377168, 4164406202270, 70677340199670, 1268718107324255, 24091289738163140, 480954355282406340, 10097484764045220626, 221918808641500960217, 5103937368681669463800

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 + 2*x^2/2! + 7*x^3/3! + 30*x^4/4! + 169*x^5/5! +...
such that A(x) = exp(-1)*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} -(-1)^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) = 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)) +-...
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) = 0.632120558828557678...
q(1) = 0.632120558828557678...
q(2) = 0.896361676485673035...
q(3) = 2.953326147114134462...
q(4) = 9.398751294399229990...
q(5) = 49.80706106045268780...
q(6) = 273.0481787175680446...
q(7) = 1956.034084445119360...
q(8) = 14858.00690837186767...
q(9) = 137211.6953065362928...
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(-1)*1 + q(0) = 1;
a(1) = exp(-1)*1 + q(1) = 1;
a(2) = exp(-1)*3 + q(2) = 2;
a(3) = exp(-1)*11 + q(3) = 7;
a(4) = exp(-1)*56 + q(4) = 30;
a(5) = exp(-1)*324 + q(5) = 169;
a(6) = exp(-1)*2324 + q(6) = 1128;
a(7) = exp(-1)*18332 + q(7) = 8700;
a(8) = exp(-1)*167544 + q(8) = 76494; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A007841, A249078, A249475, 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, -(-1)^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(-1)*P[n+1]+Q[n+1]), ", "))