A108949 - OEIS

%I #33 Jan 14 2021 02:03:29

%S 0,0,1,0,2,1,3,3,6,7,10,14,19,26,33,45,58,77,97,127,161,205,259,326,

%T 411,510,639,786,980,1197,1482,1800,2216,2677,3275,3942,4793,5749,

%U 6951,8309,9995,11912,14259,16944,20194,23926,28402,33559,39687,46767,55120,64780,76110,89222

%N Number of partitions of n with more even parts than odd parts.

%H Alois P. Heinz, <a href="/A108949/b108949.txt">Table of n, a(n) for n = 0..1000</a>

%H B. Kim, E. Kim, and J. Lovejoy, <a href="https://doi.org/10.1016/j.ejc.2020.103159">Parity bias in partitions</a>, European J. Combin., 89 (2020), 103159, 19 pp.

%F a(n) = A171966(n) - A045931(n) = A171967(n) - A108950(n). - _Reinhard Zumkeller_, Jan 21 2010

%F a(n) = Sum_{k=-floor(n/2)+(n mod 2)..-1} A240009(n,k). - _Alois P. Heinz_, Mar 30 2014

%F G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2)*(1-q^(n))/Product_{k=1..n} (1-q^(2*k))^2. - _Jeremy Lovejoy_, Jan 12 2021

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

%p with(combinat,partition):

%p evnbigrodd:=proc(n::nonnegint)

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

%p    bigcount:=0;

%p    partitions:=partition(n);

%p    for i from 1 to nops(partitions) do

%p       evencount:=0;

%p       oddcount:=0;

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

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

%p             oddcount:=oddcount+1

%p          fi;

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

%p             evencount:=evencount+1

%p          fi

%p       od;

%p       if (evencount>oddcount) then

%p          bigcount:=bigcount+1

%p       fi

%p    od;

%p    return(bigcount)

%p end proc;

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

%p # second Maple program:

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

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

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

%p     end:

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

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Mar 30 2014

%t 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 *)

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

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

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

%t 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-François Alcover_, Nov 02 2015, after _Alois P. Heinz_ *)

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

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

%Y Cf. A171966, A130780. - _Reinhard Zumkeller_, Jan 21 2010

%K nonn

%O 0,5

%A _Len Smiley_, Jul 21 2005

%E More terms from _Joerg Arndt_, Oct 04 2012