A239242 - OEIS

%I #16 Aug 30 2016 03:50:57

%S 0,1,0,1,1,1,2,1,4,2,6,3,9,5,12,9,17,14,22,22,29,33,38,48,50,68,65,95,

%T 86,128,113,172,149,226,197,295,260,379,342,485,449,613,587,773,762,

%U 967,987,1206,1269,1497,1623,1855,2063,2289,2610,2823,3280,3471

%N Number of partitions of n into distinct parts for which (number of odd parts) > (number of even parts).

%C a(n) = Sum_{k>=1} A240021(n,k). - _Alois P. Heinz_, Apr 02 2014

%H Alois P. Heinz, <a href="/A239242/b239242.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) + A239240(n) = A000009(n) for n >=1.

%e a(8) = 4 counts these partitions:  71, 53, 521, 431.

%p b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,

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

%p       b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 15 2014

%t z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];

%t Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] < Count[#, _?EvenQ] &]], {n, 0, z}] (* A239239 *)

%t Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] <= Count[#, _?EvenQ] &]], {n, 0, z}] (* A239240 *)

%t Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] == Count[#, _?EvenQ] &]], {n, 0, z}] (* A239241 *)

%t Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] > Count[#, _?EvenQ] &]], {n, 0, z}] (* A239242 *)

%t Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] >= Count[#, _?EvenQ] &]], {n, 0, z}] (* A239243 *)

%t (* _Peter J. C. Moses_, Mar 10 2014 *)

%t b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t>0, 1, 0], b[n, i-1, t]+If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Aug 30 2016, after _Alois P. Heinz_ *)

%Y Cf. A239239, A239240, A239241, A239243, A000009.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Mar 13 2014