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A045931 Number of partitions of n with equal number of even and odd parts. 60

%I #35 Jan 27 2022 20:45:29

%S 1,0,0,1,0,2,1,3,2,5,5,7,9,11,16,18,25,28,41,44,62,70,94,107,140,163,

%T 207,245,302,361,440,527,632,763,904,1090,1285,1544,1812,2173,2539,

%U 3031,3538,4202,4896,5793,6736,7934,9221,10811,12549,14661,16994,19780

%N Number of partitions of n with equal number of even and odd parts.

%C The trivariate g.f. with x marking weight (i.e., sum of the parts), t marking number of odd parts and s marking number of even parts, is 1/product((1-tx^(2j-1))(1-sx^(2j)), j=1..infinity). - _Emeric Deutsch_, Mar 30 2006

%H Alois P. Heinz, <a href="/A045931/b045931.txt">Table of n, a(n) for n = 0..3500</a> (first 1001 terms from David W. Wilson)

%F G.f.: Sum_{k>=0} x^(3*k)/Product_{i=1..k} (1-x^(2*i))^2. - _Vladeta Jovovic_, Aug 18 2007

%F a(n) = A000041(n)-A171967(n) = A130780(n)-A108950(n) = A171966(n)-A108949(n). - _Reinhard Zumkeller_, Jan 21 2010

%F a(n) = A000041(n) - A108950(n) - A108949(n) = A130780(n) + A171966(n) - A000041(n). - _Gus Wiseman_, Jan 23 2022

%e a(9) = 5 because we have [8,1], [7,2], [6,3], [5,4] and [2,2,2,1,1,1].

%e From _Gus Wiseman_, Jan 23 2022: (Start)

%e The a(0) = 1 through a(12) = 9 partitions (A = 10, empty columns indicated by dots):

%e () . . 21 . 32 2211 43 3221 54 3322 65 4332

%e 41 52 4211 63 4321 74 4431

%e 61 72 4411 83 5322

%e 81 5221 92 5421

%e 222111 6211 A1 6321

%e 322211 6411

%e 422111 7221

%e 8211

%e 22221111

%e (End)

%p g:=1/product((1-t*x^(2*j-1))*(1-s*x^(2*j)),j=1..30): gser:=simplify(series(g,x=0,56)): P[0]:=1: for n from 1 to 53 do P[n]:=subs(s=1/t,coeff(gser,x^n)) od: seq(coeff(t*P[n],t),n=0..53); # _Emeric Deutsch_, Mar 30 2006

%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 *)

%Y The version for subsets of {1..n} is A001405.

%Y Dominated by A027187 (partitions of even length).

%Y More odd/even parts: A108950/A108949.

%Y More or same number of odd/even parts: A130780/A171966.

%Y The strict case is A239241.

%Y This is column k = 0 of the triangle A240009.

%Y Counting only distinct parts gives A241638, ranked by A325700.

%Y A half-conjugate version is A277579.

%Y These partitions are ranked by A325698.

%Y A000041 counts integer partitions, strict A000009.

%Y A047993 counts balanced partitions, ranked by A106529.

%Y A257991/A257992 count odd/even parts by Heinz number.

%Y Cf. A000070, A000700, A000712, A027193, A035363, A097613, A195017.

%K nonn

%O 0,6

%A _David W. Wilson_

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)