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A108950
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Number of partitions of n with more odd parts than even parts.
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10
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1, 1, 2, 3, 4, 7, 9, 14, 18, 27, 35, 49, 64, 86, 113, 148, 192, 247, 319, 404, 517, 649, 822, 1024, 1285, 1590, 1979, 2436, 3007, 3682, 4515, 5501, 6703, 8131, 9851, 11899, 14344, 17252, 20703, 24804, 29640, 35377, 42115, 50085, 59415, 70420, 83261, 98365, 115947, 136557
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OFFSET
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1,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.
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FORMULA
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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
a(n) = A130780(n) - A045931(n) = A171967(n) - A108949(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = Sum_{k=1..n} A240009(n,k). - Alois P. Heinz, Mar 30 2014
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
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EXAMPLE
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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]}.
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MAPLE
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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);
# 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=1..80); # Alois P. Heinz, Mar 30 2014
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MATHEMATICA
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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 *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 1, 30}] (* A108950 *)
(* 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, 1, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)
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CROSSREFS
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Cf. A045931 for #even parts = #odd parts, A108949 for #even parts > #odd parts.
Cf. A171966, A171967. - Reinhard Zumkeller, Jan 21 2010
Sequence in context: A139078 A065046 A049709 * A238545 A325343 A238495
Adjacent sequences: A108947 A108948 A108949 * A108951 A108952 A108953
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KEYWORD
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nonn
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AUTHOR
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Len Smiley, Jul 21 2005
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EXTENSIONS
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More terms from Joerg Arndt, Oct 04 2012
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STATUS
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approved
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