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A045931 Number of partitions of n with equal number of even and odd parts. 21
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, 207, 245, 302, 361, 440, 527, 632, 763, 904, 1090, 1285, 1544, 1812, 2173, 2539, 3031, 3538, 4202, 4896, 5793, 6736, 7934, 9221, 10811, 12549, 14661, 16994, 19780 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..3500 (first 1001 terms from David W. Wilson)

FORMULA

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

a(n) = A000041(n)-A171967(n) = A130780(n)-A108950(n) = A171966(n)-A108949(n). - Reinhard Zumkeller, Jan 21 2010

EXAMPLE

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

MAPLE

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

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 *)

CROSSREFS

Column k=0 of A240009.

Sequence in context: A151533 A128100 A035579 * A325193 A079974 A102517

Adjacent sequences:  A045928 A045929 A045930 * A045932 A045933 A045934

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified October 19 12:29 EDT 2019. Contains 328222 sequences. (Running on oeis4.)