A299824 a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)!.

2, 22, 309, 5428, 115155, 2869242, 82187658, 2661876168, 96202473183, 3838516103310, 167606767714397, 7949901069639228, 407048805012563038, 22376916254447538882, 1314573505901491675965, 82188946843192555474704, 5448870914168179374456623, 381819805747937892412056342

Offset

1,1

Comments

For m>1, A242817(m) and a(m-1) are also the m-th and (m+1)-st terms of the sequences "Number of ways of placing X labeled balls into X unlabeled (but (m-1)-colored) boxes". For instance, sequence A144180 for 5-colored boxes (m = 6), has A144180(6) = 12880, and A144180(7) = 115155, which are A242817(6) and a(5) respectively. Same pattern can be observed for A027710, A144223, A144263 (comment added after Omar E. Pol's formula).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..368

Formula

a(n) = A189233(n+1,n). - Omar E. Pol, Feb 24 2018

a(n) ~ exp(n/LambertW(1) - 2*n) * n^(n + 1) / (sqrt(1 + LambertW(1)) * LambertW(1)^(n + 1)). - Vaclav Kotesovec, Mar 08 2018

Or: a(n) ~ (1/sqrt(1+w)) * exp(1/w-2)^n * (n/w)^(n+1), with w = LambertW(1) ~ 0.56714329... The relative error decreases from 10^-2 for a(2) to 10^-3 for a(15), but reaches 10^-3.5 only at a(45). - M. F. Hasler, Mar 09 2018

Example

a(4) = (1/e^4)*Sum_{j >= 1} j^4 * 4^j / (j-1)! = 5428.

Prog

(PARI) a(n) = round(exp(-n)*suminf(j = 1, (j^n)*(n^j)/(j-1)!)); \\ Michel Marcus, Feb 24 2018
(PARI) A299824(n, f=exp(n), S=n/f, t)=for(j=2, oo, S+=(t=j^n*n^j)/(f*=j-1); t<f&&j>n&&return(ceil(S))) \\ For n > 23, use \p## with some ## >= 2n. - M. F. Hasler, Mar 09 2018

Crossrefs

Cf. A027710, A144180, A144223, A144263, A189233, A242817, A292860.

Sequence in context: A256928 A349107 A368447 * A355724 A266888 A155674

Adjacent sequences: A299821 A299822 A299823 * A299825 A299826 A299827

Keyword

nonn

Author

Pedro Caceres, Feb 19 2018

Status

approved