A241735 - OEIS

%I #4 May 06 2014 15:04:28

%S 0,0,0,1,1,2,4,5,7,10,15,19,28,34,46,61,81,101,137,168,218,273,349,

%T 431,550,676,849,1043,1298,1579,1959,2373,2913,3526,4301,5178,6293,

%U 7544,9109,10895,13091,15591,18666,22158,26402,31269,37120,43813,51853,61027

%N 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 a part of p.

%F a(n) + A241736(n) = A000041(n) for n >= 0.

%e a(6) counts these 4 partitions:  42, 321, 2211, 21111.

%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. A241736, A241760, A241761, A000041.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Apr 28 2014