OFFSET
1,2
COMMENTS
By "proper integer partition", one means that the case {n} is excluded for having only one part, equal to the number partitioned.
The growth of this function is exponential a(n) -> c * exp(n). [This is not correct, a(n) ~ c * d^n, where d = A246828 = 2.69832910647421123126399... and c = 0.39308289517441096263558422597609193642795355676880812197435683468376... - Vaclav Kotesovec, Dec 27 2023]
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..2320 (first 70 terms from Vincenzo Librandi)
FORMULA
a(n) = sum( sum( a(i), i in p) , p in P*(n)) where P*(n) is the set of all integer partitions of n excluding {n}, p is a partition of P*(n), i is a part of p.
a(n) ~ exp(k) * a(n-1), k = 0.992632731... (conjecture). - Bill McEachen, Dec 26 2023
EXAMPLE
a(4) = (a(3)+a(1))+(a(2)+a(2))+(a(2)+a(1)+a(1))+(a(1)+a(1)+a(1)+a(1)) = (6 + 1) + (2 + 2) + (2 + 2*1) + (4*1) = 7 + 4 + 4 + 4 = 19.
MAPLE
b:= proc(n, i) option remember; `if`(n<2, [1, n], `if`(i<1, 0,
b(n, i-1)+(p-> p+[0, p[1]*a(i)])(b(n-i, min(n-i, i)))))
end:
a:= n-> b(n, n-1)[2]:
seq(a(n), n=1..33); # Alois P. Heinz, Dec 27 2023
MATHEMATICA
Clear[a]; a[1] := 1; a[n_Integer] :=
a[n] = Plus @@ Map[Function[p, Plus @@ Map[a, p]], Drop[IntegerPartitions[n], 1]]; Table[ a[n], {n, 1, 30}]
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Olivier Gérard, Jul 30 2012
STATUS
approved