login
A241736
Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) is not a part of p.
4
1, 1, 2, 2, 4, 5, 7, 10, 15, 20, 27, 37, 49, 67, 89, 115, 150, 196, 248, 322, 409, 519, 653, 824, 1025, 1282, 1587, 1967, 2420, 2986, 3645, 4469, 5436, 6617, 8009, 9705, 11684, 14093, 16906, 20290, 24247, 28992, 34508, 41103, 48773, 57865, 68438, 80941
OFFSET
0,3
COMMENTS
The partition {n} is included in the count.
FORMULA
a(n) + A241735(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 7 partitions: 6, 51, 411, 33, 3111, 222, 11111.
MATHEMATICA
z = 55; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]; g1[p_] := Min[-Differences[p]];
Table[Count[f[n], p_ /; MemberQ[p, g[p]]], {n, 0, z}] (* A241735 *)
Table[Count[f[n], p_ /; ! MemberQ[p, g[p]]], {n, 0, z}] (* A241736 *)
Table[Count[f[n], p_ /; MemberQ[p, g1[p]]], {n, 0, z}] (* A241760 *)
Table[Count[f[n], p_ /; ! MemberQ[p, g1[p]]], {n, 0, z}](* A241761 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 28 2014
STATUS
approved