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A108949 Number of partitions of n with more even parts than odd parts. 8
0, 0, 1, 0, 2, 1, 3, 3, 6, 7, 10, 14, 19, 26, 33, 45, 58, 77, 97, 127, 161, 205, 259, 326, 411, 510, 639, 786, 980, 1197, 1482, 1800, 2216, 2677, 3275, 3942, 4793, 5749, 6951, 8309, 9995, 11912, 14259, 16944, 20194, 23926, 28402, 33559, 39687, 46767, 55120, 64780, 76110, 89222 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n) = A171966(n) - A045931(n) = A171967(n) - A108950(n). - Reinhard Zumkeller, Jan 21 2010

a(n) = Sum_{k=-floor(n/2)+(n mod 2)..-1} A240009(n,k). - Alois P. Heinz, Mar 30 2014

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

EXAMPLE

a(6) = 3: {[6], [4,2], [2,2,2]}; a(7) = 3: {[4,2,1], [3,2,2], [2,2,2,1]}.

MAPLE

with(combinat, partition):

evnbigrodd:=proc(n::nonnegint)

   local evencount, oddcount, bigcount, parts, i, j;

   bigcount:=0;

   partitions:=partition(n);

   for i from 1 to nops(partitions) do

      evencount:=0;

      oddcount:=0;

      for j from 1 to nops(partitions[i]) do

         if (op(j, partitions[i]) mod 2 <>0) then

            oddcount:=oddcount+1

         fi;

         if (op(j, partitions[i]) mod 2 =0) then

            evencount:=evencount+1

         fi

      od;

      if (evencount>oddcount) then

         bigcount:=bigcount+1

      fi

   od;

   return(bigcount)

end proc;

seq(evnbigrodd(i), i=1..42);

# second Maple program:

b:= proc(n, i, t) option remember; `if`(n=0,

      `if`(t<0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+

      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))

    end:

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

seq(a(n), n=0..80);  # Alois P. Heinz, Mar 30 2014

MATHEMATICA

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, _?OddQ] == Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)

TableForm[t] (* partitions, vertical format *)

Table[Length[p[n]], {n, 0, 30}] (* A045931 *)

(* Peter J. C. Moses, Mar 10 2014 *)

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t<0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-Fran├žois Alcover, Nov 02 2015, after Alois P. Heinz *)

PROG

(PARI) a(n) = {nb = 0; forpart(p=n, nb += (2*#(select(x->x%2, Vec(p))) < #p); ); nb; } \\ Michel Marcus, Nov 02 2015

CROSSREFS

Cf. A045931 for #even parts = #odd parts, A108950 for #even parts < #odd parts.

Cf. A171966, A130780. - Reinhard Zumkeller, Jan 21 2010

Sequence in context: A096373 A216961 A241379 * A167704 A109522 A052959

Adjacent sequences:  A108946 A108947 A108948 * A108950 A108951 A108952

KEYWORD

nonn

AUTHOR

Len Smiley, Jul 21 2005

EXTENSIONS

More terms from Joerg Arndt, Oct 04 2012

STATUS

approved

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Last modified August 24 00:29 EDT 2017. Contains 291052 sequences.