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
EXAMPLE
a(9) = 5 because we have [8,1], [7,2], [6,3], [5,4] and [2,2,2,1,1,1].
From Gus Wiseman, Jan 23 2022: (Start)
The a(0) = 1 through a(12) = 9 partitions (A = 10, empty columns indicated by dots):
() . . 21 . 32 2211 43 3221 54 3322 65 4332
41 52 4211 63 4321 74 4431
61 72 4411 83 5322
81 5221 92 5421
222111 6211 A1 6321
322211 6411
422111 7221
8211
22221111
(End)
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
KEYWORD
nonn
AUTHOR
STATUS
approved