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%I #5 May 06 2014 15:06:17
%S 0,0,0,1,0,0,2,1,2,2,3,2,6,5,8,9,11,11,17,17,23,26,32,36,47,51,63,71,
%T 84,93,115,127,151,171,201,225,266,297,346,389,448,501,580,648,740,
%U 831,945,1056,1201,1340,1517,1695,1910,2130,2399,2670
%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 a part of p.
%F a(n) + A241761(n) = A000041(n) for n >= 0.
%e a(6) counts these 2 partitions: 42, 321.
%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 0,7
%A _Clark Kimberling_, Apr 28 2014