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A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

%I A240057 #14 Aug 30 2016 10:31:52
%S A240057 0,2,3,4,6,10,14,21,28,40,53,74,97,131,171,225,290,377,480,616,779,
%T A240057 987,1238,1556,1935,2411,2981,3685,4527,5562,6793,8295,10081,12241,
%U A240057 14805,17890,21538,25906,31062,37201,44429,53004,63070,74964,88898,105297
%N A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).
%C A240057 Let # denote "number of" and c(p) = conjugate of partitionp.  Then
%C A240057 A240057(n) = # p such that min(p) not = max(c(p));
%C A240057 A039899(n) = # p such that min(p) < max(c(p));
%C A240057 A039900(n) = # p such that min(p) <= max(c(p));
%C A240057 A006141(n) = # p such that min(p) = max(c(p));
%C A240057 A003114(n) = # p such that min(p) > max(c(p));
%C A240057 A003016(n) = # p such that min(p) >= max(c(p));
%C A240057 A064173(n) = # p such that max(p) < max(c(p));
%C A240057 A064174(n) = # p such that max(p) <= max(c(p));
%C A240057 A047993(n) = # p such that max(p) = max(c(p)).
%C A240057 See A240178 for related sequences. - _Clark Kimberling_, Apr 11 2014
%H A240057 Alois P. Heinz, <a href="/A240057/b240057.txt">Table of n, a(n) for n = 1..1000</a>
%F A240057 a(n) + A006141(n) = A000041(n) for n > 0.
%e A240057 a(9) = 28 counts all the 30 partitions of 9 except 333 and 2211111.
%p A240057 b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A240057       `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))
%p A240057     end:
%p A240057 a:= n->combinat[numbpart](n)-add(b(n-j^2, j-1), j=0..isqrt(n)):
%p A240057 seq(a(n), n=1..50);  # _Alois P. Heinz_, Apr 03 2014
%t A240057 z = 60; f[n_] := f[n] = IntegerPartitions[n];
%t A240057 t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Max[p]]], {n, 0, z}]  (* A003106 *)
%t A240057 t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Max[p]]], {n, 0, z}] (* A003114 *)
%t A240057 t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Max[p]]], {n, 0, z}] (* A006141 *)
%t A240057 tt = Table[Count[f[n], p_ /; Max[p] != Count[p, Max[p]]], {n, 0, z}] (* A240057 *)
%t A240057 t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Max[p]]], {n, 0, z}] (* A039899 *)
%t A240057 t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Max[p]]], {n, 0, z}] (* A039900 *)
%t A240057 (* second program: *)
%t A240057 b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]];
%t A240057 a[n_] := PartitionsP[n] - Sum[b[n - j^2, j - 1], {j, 0, Sqrt[n]}];
%t A240057 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Aug 30 2016, after _Alois P. Heinz_ *)
%Y A240057 Cf. A003106, A003114, A006141, A039899, A039900.
%K A240057 nonn,easy
%O A240057 1,2
%A A240057 _Clark Kimberling_, Apr 02 2014