A240056 - OEIS

%I #12 Jan 15 2022 03:27:23

%S 0,1,1,1,3,4,5,9,12,16,24,32,42,59,77,100,134,173,221,288,366,463,590,

%T 741,926,1163,1444,1787,2215,2726,3342,4101,5003,6087,7402,8964,10827,

%U 13069,15718,18865,22617,27041,32263,38453,45719,54264,64326,76102,89884

%N Number of partitions of n such that m(1) > m(2), where m = multiplicity.

%H Vaclav Kotesovec, <a href="/A240056/b240056.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%F a(n) = A000041(n) - A182713(n+2) = a(n+1) - A174455(n) for n >= 0.

%F a(n) ~ exp(sqrt(2*n/3)*Pi) / (2 * 3^(3/2) * n). - _Vaclav Kotesovec_, Jan 15 2022

%e a(7) counts these 9 partitions: 61, 511, 4111, 331, 3211, 31111, 22111, 211111, 1111111.

%t z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}]  (* A182713 *)

%t t2 = Table[Count[f[n], p_ /; Count[p, 1] <= Count[p, 2]], {n, 0, z}] (* A182713(n+2) *)

%t t3 = Table[Count[f[n], p_ /; Count[p, 1] == Count[p, 2]], {n, 0, z}] (* A174455 *)

%t t4 = Table[Count[f[n], p_ /; Count[p, 1] > Count[p, 2]], {n, 0, z}]  (* A240056 *)

%t t5 = Table[Count[f[n], p_ /; Count[p, 1] >= Count[p, 2]], {n, 0, z}] (* A240056(n+1) *)

%Y Cf. A182713, A174455, A000041.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Mar 31 2014