A240057 - OEIS

%I #14 Aug 30 2016 10:31:52

%S 0,2,3,4,6,10,14,21,28,40,53,74,97,131,171,225,290,377,480,616,779,

%T 987,1238,1556,1935,2411,2981,3685,4527,5562,6793,8295,10081,12241,

%U 14805,17890,21538,25906,31062,37201,44429,53004,63070,74964,88898,105297

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

%C Let # denote "number of" and c(p) = conjugate of partitionp.  Then

%C A240057(n) = # p such that min(p) not = max(c(p));

%C A039899(n) = # p such that min(p) < max(c(p));

%C A039900(n) = # p such that min(p) <= max(c(p));

%C A006141(n) = # p such that min(p) = max(c(p));

%C A003114(n) = # p such that min(p) > max(c(p));

%C A003016(n) = # p such that min(p) >= max(c(p));

%C A064173(n) = # p such that max(p) < max(c(p));

%C A064174(n) = # p such that max(p) <= max(c(p));

%C A047993(n) = # p such that max(p) = max(c(p)).

%C See A240178 for related sequences. - _Clark Kimberling_, Apr 11 2014

%H Alois P. Heinz, <a href="/A240057/b240057.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) + A006141(n) = A000041(n) for n > 0.

%e a(9) = 28 counts all the 30 partitions of 9 except 333 and 2211111.

%p b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,

%p       `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))

%p     end:

%p a:= n->combinat[numbpart](n)-add(b(n-j^2, j-1), j=0..isqrt(n)):

%p seq(a(n), n=1..50);  # _Alois P. Heinz_, Apr 03 2014

%t z = 60; f[n_] := f[n] = IntegerPartitions[n];

%t t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Max[p]]], {n, 0, z}]  (* A003106 *)

%t t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Max[p]]], {n, 0, z}] (* A003114 *)

%t t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Max[p]]], {n, 0, z}] (* A006141 *)

%t tt = Table[Count[f[n], p_ /; Max[p] != Count[p, Max[p]]], {n, 0, z}] (* A240057 *)

%t t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Max[p]]], {n, 0, z}] (* A039899 *)

%t t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Max[p]]], {n, 0, z}] (* A039900 *)

%t (* second program: *)

%t 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 a[n_] := PartitionsP[n] - Sum[b[n - j^2, j - 1], {j, 0, Sqrt[n]}];

%t Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Aug 30 2016, after _Alois P. Heinz_ *)

%Y Cf. A003106, A003114, A006141, A039899, A039900.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Apr 02 2014