# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a240056 Showing 1-1 of 1 %I A240056 #12 Jan 15 2022 03:27:23 %S A240056 0,1,1,1,3,4,5,9,12,16,24,32,42,59,77,100,134,173,221,288,366,463,590, %T A240056 741,926,1163,1444,1787,2215,2726,3342,4101,5003,6087,7402,8964,10827, %U A240056 13069,15718,18865,22617,27041,32263,38453,45719,54264,64326,76102,89884 %N A240056 Number of partitions of n such that m(1) > m(2), where m = multiplicity. %H A240056 Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz) %F A240056 a(n) = A000041(n) - A182713(n+2) = a(n+1) - A174455(n) for n >= 0. %F A240056 a(n) ~ exp(sqrt(2*n/3)*Pi) / (2 * 3^(3/2) * n). - _Vaclav Kotesovec_, Jan 15 2022 %e A240056 a(7) counts these 9 partitions: 61, 511, 4111, 331, 3211, 31111, 22111, 211111, 1111111. %t A240056 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 A240056 t2 = Table[Count[f[n], p_ /; Count[p, 1] <= Count[p, 2]], {n, 0, z}] (* A182713(n+2) *) %t A240056 t3 = Table[Count[f[n], p_ /; Count[p, 1] == Count[p, 2]], {n, 0, z}] (* A174455 *) %t A240056 t4 = Table[Count[f[n], p_ /; Count[p, 1] > Count[p, 2]], {n, 0, z}] (* A240056 *) %t A240056 t5 = Table[Count[f[n], p_ /; Count[p, 1] >= Count[p, 2]], {n, 0, z}] (* A240056(n+1) *) %Y A240056 Cf. A182713, A174455, A000041. %K A240056 nonn,easy %O A240056 0,5 %A A240056 _Clark Kimberling_, Mar 31 2014 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE