A249476 E.g.f.: exp(3)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 3^n/Product_{k=1..n} (k - x^k).

1, 1, 6, 35, 242, 1773, 15056, 136652, 1393722, 15257919, 183206388, 2347929936, 32602306542, 479885400177, 7563888117504, 125952344438838, 2225653414414386, 41351620513521627, 810520833521436732, 16633643598838880244, 358221783030360367014, 8051927483267030640573

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 + 6*x^2/2! + 35*x^3/3! + 242*x^4/4! + 1773*x^5/5! +...
such that A(x) = exp(3)*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} 3^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) = 3/(1-x) + 3^2/((1-x)*(2-x^2)) + 3^3/((1-x)*(2-x^2)*(3-x^3)) + 3^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 3^5/((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) = 19.085536923187667740...
q(1) = 19.085536923187667740...
q(2) = 54.256610769563003222...
q(3) = 185.94090615506434515...
q(4) = 882.79006769850939349...
q(5) = 4734.7139631128043480...
q(6) = 31622.787809488139829...
q(7) = 231556.06287587632502...
q(8) = 1971489.1982585546039...
q(9) = 18370572.391163877342...
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(3)*1 - q(0) = 1;
a(1) = exp(3)*1 - q(1) = 1;
a(2) = exp(3)*3 - q(2) = 6;
a(3) = exp(3)*11 - q(3) = 35;
a(4) = exp(3)*56 - q(4) = 242;
a(5) = exp(3)*324 - q(5) = 1773;
a(6) = exp(3)*2324 - q(6) = 15056;
a(7) = exp(3)*18332 - q(7) = 136652;
a(8) = exp(3)*167544 - q(8) = 1393722; ...
		

Crossrefs

Cf. A007841, A249078, A249474, A249475, 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, 3^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(3)*P[n+1]-Q[n+1]), ", "))