A241761 - OEIS

%I #4 May 06 2014 15:04:05

%S 1,1,2,2,5,7,9,14,20,28,39,54,71,96,127,167,220,286,368,473,604,766,

%T 970,1219,1528,1907,2373,2939,3634,4472,5489,6715,8198,9972,12109,

%U 14658,17711,21340,25669,30796,36890,44082,52594,62613,74435,88303,104613,123698

%N Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >=  ... >= x(k), of n such that min(x(i) - x(i-1))  is not a part of p.

%C The partition {n} is included in the count.

%F a(n) + A241760(n) = A000041(n) for n >= 0.

%e a(6) counts these 6 partitions:  6, 51, 411, 33, 3111, 222, 2211, 21111, 111111.

%t z = 55; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]];  g1[p_] := Min[-Differences[p]];

%t Table[Count[f[n], p_ /; MemberQ[p, g[p]]], {n, 0, z}]   (* A241735 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, g[p]]], {n, 0, z}] (* A241736 *)

%t Table[Count[f[n], p_ /; MemberQ[p, g1[p]]], {n, 0, z}]  (* A241760 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, g1[p]]], {n, 0, z}](* A241761 *)

%Y Cf. A241735, A241736, A241761, A000041.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 28 2014