OFFSET
0,4
COMMENTS
Perhaps the largest possible number of ways of choosing (v1, v2, ..., vn), possibly with repetition, from {b1, b2, ..., bn} with b1 < b2 < ... < bn, such that v1 + v2 + ... + vn < b1 + b2 + ... + bn. Clearly the actual number of ways depends on the particular values of {b1, b2, ..., bn}, but {1, n, n^2, ..., n^(n-1)} produces this result for the number of sums strictly less than (n^n-1)/(n-1) = A023037(n).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..386
FORMULA
Limit_{n->oo} a(n)/(e*a(n-1)) - n = -1/2.
Limit_{n->oo} a(n)/n^n = 1/(e-1).
EXAMPLE
a(4) = 4!*(0^1/1! + 1^2/2! + 2^3/3! + 3^4/4!) = 0 + 12 + 32 + 81 = 125.
MATHEMATICA
Table[n! Sum[(k-1)^k/k!, {k, n}], {n, 0, 17}] (* Stefano Spezia, Sep 11 2022 *)
PROG
(PARI) a(n) = n!*sum(k=1, n, (k-1)^k/k!); \\ Seiichi Manyama, Jul 15 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, Oct 14 2002
STATUS
approved