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A238217
The total number of 2's in all partitions of n into an even number of distinct parts.
2
0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 11, 14, 16, 19, 22, 26, 30, 35, 41, 47, 55, 63, 73, 84, 97, 110, 127, 145, 166, 188, 215, 243, 277, 313, 354, 400, 452, 508, 573, 644, 723, 811, 910, 1018, 1139, 1273, 1421, 1586, 1768, 1968, 2190, 2436
OFFSET
0,11
COMMENTS
The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).
LINKS
FORMULA
a(n) = Sum_{j=1..round(n/4)} A067659(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067661(n-4*j).
G.f.: (1/2)*(x^2/(1+x^2))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^2/(1-x^2))*(Product_{n>=1} 1 - x^n).
EXAMPLE
a(12) = 3 because the partitions in question are: 10+2, 6+3+2+1, 5+4+2+1.
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&Length[#] == Length[ Union[#]]&&MemberQ[#, 2]&]], {n, 0, 50}] (* Harvey P. Dale, Dec 09 2014 *)
PROG
(PARI) seq(n)={my(A=O(x^(n-1))); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x^2)) - eta(x + A)/(1-x^2))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020
CROSSREFS
Column k=2 of A238451.
Sequence in context: A265771 A239513 A029018 * A185226 A096765 A025147
KEYWORD
nonn
AUTHOR
Mircea Merca, Feb 20 2014
EXTENSIONS
Terms a(51) and beyond from Andrew Howroyd, May 01 2020
STATUS
approved