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
KEYWORD
nonn
AUTHOR
Pedro Caceres, Feb 19 2018
STATUS
approved