OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)
EXAMPLE
a(4) = 4 because of the 5 unrestricted partitions of 4, only one, 3+1 uses each of its summands just once and 1,1 is not an increasing sequence.
From Gus Wiseman, Jan 23 2019: (Start)
The a(1) = 1 through a(8) = 11 integer partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (311) (33) (322) (44)
(211) (2111) (222) (511) (422)
(1111) (11111) (411) (4111) (611)
(3111) (22111) (2222)
(21111) (31111) (5111)
(111111) (211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
(End)
MAPLE
b:= proc(n, i, t) option remember;
if n<0 then 0
elif n=0 then 1
elif i=1 then `if`(n>t, 1, 0)
elif i=0 then 0
else b(n, i-1, t)
+add(b(n-i*j, i-1, j), j=t+1..floor(n/i))
fi
end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 21 2011
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n==0, 1, i==1, If[n>t, 1, 0], i == 0, 0 , True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t+1, Floor[n/i]}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], OrderedQ@*Split]], {n, 20}] (* Gus Wiseman, Jan 23 2019 *)
PROG
(Haskell)
a100471 n = p 0 (n + 1) 1 n where
p m m' k x | x == 0 = if m < m' || m == 0 then 1 else 0
| x < k = 0
| m == 0 = p 1 m' k (x - k) + p 0 m' (k + 1) x
| otherwise = p (m + 1) m' k (x - k) +
if m < m' then p 0 m (k + 1) x else 0
-- Reinhard Zumkeller, Dec 27 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
David S. Newman, Nov 21 2004
EXTENSIONS
Corrected and extended by Vladeta Jovovic, Nov 24 2004
Name edited by Gus Wiseman, Jan 23 2019
STATUS
approved