A243806 G.f.: exp( Integral Sum_{n>=1} (n+1)!*x^(n-1) / Product_{k=1..n} (1-k*x) dx ).

1, 2, 6, 24, 123, 786, 6112, 56504, 607833, 7467066, 103120674, 1580416008, 26598861595, 487397118314, 9654657563748, 205508121922824, 4676833854264603, 113293413849825702, 2910255267932697130, 79008535866112062440, 2260164362082172615833, 67947314033110789074486

Offset

0,2

Comments

Compare g.f. to:

exp( Integral Sum_{n>=1} (n+1)!*x^(n-1) / Product_{k=1..n} (1+k*x) dx ) = 1/(1-2*x).

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: exp( Sum_{n>=1} A005649(n)*x^n/n ) where Sum_{n>=0} A005649(n)*x^n/n! = 1/(2-exp(x))^2.

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 123*x^4 + 786*x^5 + 6112*x^6 +...
where the logarithmic derivative is given by the series:
A'(x)/A(x) = 2!/(1-x) + 3!*x/((1-x)*(1-2*x)) + 4!*x^2/((1-x)*(1-2*x)*(1-3*x)) + 5!*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + 6!*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) +...
Explicitly,
A'(x)/A(x) = 2 + 8*x + 44*x^2 + 308*x^3 + 2612*x^4 + 25988*x^5 + 296564*x^6 +...+ A005649(n+1)*x^n +...
compare to:
1/(2-exp(x))^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 308*x^4/4! + 2612*x^5/5! + 25988*x^6/6! +...+ A005649(n)*x^n/n! +...

Prog

(PARI) {a(n)=local(A=1+x); A=exp(intformal(sum(m=1, n+1, (m+1)!*x^(m-1)/prod(k=1, m, 1-k*x +x*O(x^n))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From g.f. exp( Sum_{n>=1} A005649(n)*x^n/n ): */
{A005649(n)=n!*polcoeff(1/(2-exp(x +x*O(x^n)))^2, n)}
{a(n)=polcoeff(exp(sum(m=1, n, A005649(n)*x^n/n) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A005649.

Sequence in context: A104983 A144251 A304198 * A201158 A356634 A191343

Adjacent sequences: A243803 A243804 A243805 * A243807 A243808 A243809

Keyword

nonn

Author

Paul D. Hanna, Jun 11 2014

Status

approved