|
%I #33 Aug 19 2021 20:06:44
%S 1,1,2,3,4,7,9,14,18,27,35,49,64,86,113,148,192,247,319,404,517,649,
%T 822,1024,1285,1590,1979,2436,3007,3682,4515,5501,6703,8131,9851,
%U 11899,14344,17252,20703,24804,29640,35377,42115,50085,59415,70420,83261,98365,115947,136557
%N Number of partitions of n with more odd parts than even parts.
%H Alois P. Heinz, <a href="/A108950/b108950.txt">Table of n, a(n) for n = 1..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 G.f.: Sum_{k>=0} x^k*(1-x^(2*k))/Product_{i=1..k} (1-x^(2*i))^2. - _Vladeta Jovovic_, Aug 19 2007
%F a(n) = A130780(n) - A045931(n) = A171967(n) - A108949(n). - _Reinhard Zumkeller_, Jan 21 2010
%F a(n) = Sum_{k=1..n} 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-n)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2. - _Jeremy Lovejoy_, Jan 12 2021
%e a(4) = 3: {[3,1], [2,1,1], [1,1,1,1]}; a(5) = 4: {[5], [3,1,1], [2,1,1,1], [1,1,1,1,1]}.
%p with(combinat,partition):oddbigrevn:=proc(n::nonnegint) local evencount,oddcount,bigcount,parts,i,j; printlevel:=-1;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; printlevel:=1; return(bigcount) end proc; seq(oddbigrevn(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=1..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, 15}] (* partitions of n with # odd parts > # even parts *)
%t TableForm[t] (* partitions, vertical format *)
%t Table[Length[p[n]], {n, 1, 30}] (* A108950 *)
%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, 1, 80}] (* _Jean-François Alcover_, Nov 16 2015, after _Alois P. Heinz_ *)
%Y Cf. A045931 for #even parts = #odd parts, A108949 for #even parts > #odd parts.
%Y Cf. A171966, A171967. - _Reinhard Zumkeller_, Jan 21 2010
%K nonn
%O 1,3
%A _Len Smiley_, Jul 21 2005
%E More terms from _Joerg Arndt_, Oct 04 2012
|
