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A293422
The PDO_t(n) function (Number of tagged parts over all the partitions of n with designated summands in which all parts are odd).
2
1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 196, 264, 352, 468, 614, 800, 1036, 1332, 1704, 2168, 2744, 3456, 4331, 5408, 6724, 8328, 10278, 12640, 15496, 18936, 23072, 28030, 33960, 41040, 49470, 59488, 71368, 85428, 102042, 121632, 144692, 171792, 203584
OFFSET
1,2
LINKS
FORMULA
G.f.: q * Product_{k>0} ((1 - q^(2*k))*(1 - q^(3*k))^2*(1 - q^(12*k))^2)/((1 - q^k)^2*(1 - q^(6*k))).
a(n) ~ exp(sqrt(5*n)*Pi/3) / (3 * 2^(3/2) * 5^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 15 2017
EXAMPLE
n = 4 n = 5 n = 6
------------------- ----------------------- ---------------------------
3'+ 1' -> 2 5' -> 1 5'+ 1' -> 2
1'+ 1 + 1 + 1 -> 1 3'+ 1'+ 1 -> 2 3'+ 3 -> 1
1 + 1'+ 1 + 1 -> 1 3'+ 1 + 1' -> 2 3 + 3' -> 1
1 + 1 + 1'+ 1 -> 1 1'+ 1 + 1 + 1 + 1 -> 1 3'+ 1'+ 1 + 1 -> 2
1 + 1 + 1 + 1'-> 1 1 + 1'+ 1 + 1 + 1 -> 1 3'+ 1 + 1'+ 1 -> 2
1 + 1 + 1'+ 1 + 1 -> 1 3'+ 1 + 1 + 1' -> 2
1 + 1 + 1 + 1'+ 1 -> 1 1'+ 1 + 1 + 1 + 1 + 1 -> 1
1 + 1 + 1 + 1 + 1'-> 1 1 + 1'+ 1 + 1 + 1 + 1 -> 1
1 + 1 + 1'+ 1 + 1 + 1 -> 1
1 + 1 + 1 + 1'+ 1 + 1 -> 1
1 + 1 + 1 + 1 + 1'+ 1 -> 1
1 + 1 + 1 + 1 + 1 + 1'-> 1
------------------- ----------------------- ---------------------------
a(4) = 6. a(5) = 10. a(6) = 16.
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(3*k))^2 * (1-x^(12*k))^2 / ((1-x^k)^2 * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 15 2017 *)
PROG
(Ruby)
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(n)
partition(n, 1, n).select{|i| i.all?{|j| j.odd?}}.map{|a| a.each_with_object(Hash.new(0)){|v, o| o[v] += 1}.values}.map{|i| i.size * i.inject(:*)}.inject(:+)
end
def A293422(n)
(1..n).map{|i| A(i)}
end
p A293422(40)
CROSSREFS
Cf. A102186 (PDO(n)), A293421.
Sequence in context: A241903 A261204 A374145 * A132002 A098151 A137414
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 08 2017
STATUS
approved