A321016 Triangle read by rows: number of partitions of n into distinct and consecutive parts with largest part k (n >= 1, 1 <= k <= n)..

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset

1

Comments

Suggested by A321440.

Links

Table of n, a(n) for n=1..120.

Example

Triangle begins:
[1]
[0, 1]
[0, 1, 1]
[0, 0, 0, 1]
[0, 0, 1, 0, 1]
[0, 0, 1, 0, 0, 1]
[0, 0, 0, 1, 0, 0, 1]
[0, 0, 0, 0, 0, 0, 0, 1]
[0, 0, 0, 1, 1, 0, 0, 0, 1]
[0, 0, 0, 1, 0, 0, 0, 0, 0, 1]
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1]
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1]
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
[0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1]
...

Maple

with(combinat);
f2:=proc(n) local i, j, p, t0, t1, t2;
t0:=Array(1..n, 0); t1:=partition(n); p:=numbpart(n);
for i from 1 to p do t2:=t1[i];
if nops(convert(t2, set))=nops(t2) then
# now have a partition t2 of n into distinct parts
gd:=1;
  if nops(t2)>1 then
  for j from 1 to nops(t2)-1 do if t2[j+1]-t2[j]>1 then gd:=-1; fi;
  od; fi;
# now have a partition t2 of n into distinct consecutive parts
if gd=1 then t0[t2[-1]]:=t0[t2[-1]]+1; fi;
fi;
                 od:
[seq(t0[j], j=1..n)];
end proc;
for n from 1 to 15 do lprint(f2(n)) od:

Crossrefs

Row sums are A001227.

Cf. A321440.

Sequence in context: A077052 A133566 A185907 * A077051 A115955 A106344

Adjacent sequences: A321013 A321014 A321015 * A321017 A321018 A321019

Keyword

nonn,tabl

Author

N. J. A. Sloane, Nov 09 2018

Status

approved