A232688 Expansion of 1/(1-x)^(1+2*x).

1, 1, 6, 24, 160, 1100, 9168, 84336, 868544, 9806688, 120754080, 1608251040, 23036978688, 353070167424, 5764504370688, 99875064159360, 1830107743518720, 35360088597841920, 718450398365755392, 15313427783968370688, 341652563064038062080, 7962756873168407869440

Offset

0,3

Comments

Generally, if E.g.f. = 1/(1-x)^(1+p*x) then a(n) ~ n! * n^p/p! * (1 - p^2*log(n)/n). - Vaclav Kotesovec, Nov 29 2013

Links

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

Formula

E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} (k + 2*x).

a(n) ~ n! * n^2/2 * (1-4*log(n)/n). - Vaclav Kotesovec, Nov 29 2013

Example

E.g.f.: A(x) = 1 + x + 6*x^2/2! + 24*x^3/3! + 160*x^4/4! + 1100*x^5/5! + 9168*x^6/6! +...
where
A(x) = 1 + (1+2*x)*x + (1+2*x)*(2+2*x)*x^2/2! + (1+2*x)*(2+2*x)*(3+2*x)*x^3/3! + (1+2*x)*(2+2*x)*(3+2*x)*(4+2*x)*x^4/4! + (1+2*x)*(2+2*x)*(3+2*x)*(4+2*x)*(5+2*x)*x^5/5! +...

Mathematica

CoefficientList[Series[1/(1-x)^(1+2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 29 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(sum(m=0, n, prod(k=1, m, k+2*x)*x^m/m!)+x*O(x^n), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff((1-x+x*O(x^n))^(-1-2*x), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A073479.

Sequence in context: A165491 A165638 A122829 * A221980 A359693 A225827

Adjacent sequences: A232685 A232686 A232687 * A232689 A232690 A232691

Keyword

nonn

Author

Paul D. Hanna, Nov 27 2013

Status

approved