A090205 a(n) = n^n * (n+1)^(n+1).

1, 4, 108, 6912, 800000, 145800000, 38423222208, 13816758796288, 6499837226778624, 3874204890000000000, 2853116706110000000000, 2543867514227405321404416, 2700464873652770753916960768, 3365550251744726983606907650048, 4865879898782263021500000000000000

Offset

0,2

Comments

I was made aware of this sequence by M. Frazer.

These are the denominators of the rapidly converging approximation to e given by (n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n (see A111130). - Harlan J. Brothers, Aug 25 2023

Links

Table of n, a(n) for n=0..14.

H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, 20 (1998), 25-29.

Formula

a(n) = A000312(n) * A000312(n+1).

a(n) ~ e^(1 + 1/(2*n)) * n^(2*n+1). - Harlan J. Brothers, Aug 25 2023

Example

a(4) = 4^4 * 5^5 = 256 * 3125 = 800000.

Mathematica

Table[n^n (n+1)^(n+1), {n, 20}] (* Harvey P. Dale, May 29 2016 *)
Denominator[Table[(n + 2)^(n + 2)/(n + 1)^(n + 1) - (n + 1)^(n + 1)/n^n, {n, 20}]] (* Harlan J. Brothers, Aug 25 2023 *)
Join[{1}, Times@@@Partition[Table[n^n, {n, 20}], 2, 1]] (* Harvey P. Dale, Feb 04 2024 *)

Prog

(Maxima) A090205[n]:=n^n*(n+1)^(n+1)$ makelist(A090205[n], n, 1, 30); /* Martin Ettl, Oct 29 2012 */

Crossrefs

Cf. A000312, A111130.

Sequence in context: A360092 A123056 A024263 * A061464 A107048 A269187

Adjacent sequences: A090202 A090203 A090204 * A090206 A090207 A090208

Keyword

easy,nonn

Author

Philip Newton, Jan 22 2004

Extensions

More terms from Harlan J. Brothers, Aug 25 2023

Status

approved