0,3

Comment from John MCKAY (mckay(AT)encs.concordia.ca), Mar 06 2009 (Start): Apparently a(n) = number of partitions (p_1, p_2, ..., p_k) of n+1, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i > p_{i+1}+...+p_k.

For example, the five partitions of 4, written in nonincreasing order, are [1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4]. Only the last two satisfy the condition, and a(3)=2. The Maple program below verifies this for small values of n. (End)

(Maple code from John McKay) with(combinat); N:=8; a:=array(1..N); c:=array(1..N);

for n from 1 to N do p:=partition(n); np:=nops(p); t:=0;

for s to np do r:=p[s]; r:=sort(r, `>`); nr:=nops(r); j:=1;

while j<nr and r[j]>sum(r[k], k=j+1..nr) do j:=j+1; od; # gives A040039

#while j<nr and r[j]>= sum(r[k], k=j+1..nr) do j:=j+1; od; # gives A018819

if j=nr then t:=t+1; fi od; a[n]:=t; od;

Cf. A000123, A018819.

Cf. A018819, A088567, A089054.

Sequence in context: A085885 A064986 A029019 this_sequence A008667 A109763 A119620

Adjacent sequences: A040036 A040037 A040038 this_sequence A040040 A040041 A040042

nonn,easy,more

N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)