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A238218
The total number of 3's in all partitions of n into an even number of distinct parts.
2
0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 17, 20, 24, 27, 32, 38, 43, 50, 59, 67, 77, 90, 102, 117, 135, 153, 175, 200, 226, 257, 292, 330, 373, 422, 475, 535, 603, 677, 760, 853, 955, 1069, 1196, 1336, 1491, 1663, 1853, 2063, 2295
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/6)} A067659(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067661(n-6*j).
G.f.: (1/2)*(x^3/(1+x^3))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^3/(1-x^3))*(Product_{n>=1} 1 - x^n).
EXAMPLE
a(13) = 3 because the partitions in question are: 10+3, 7+3+2+1, 5+4+3+1.
PROG
(PARI) seq(n)={my(A=O(x^(n-2))); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x^3)) - eta(x + A)/(1-x^3))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020
CROSSREFS
Column k=3 of A238451.
Sequence in context: A309689 A029049 A094983 * A015744 A118301 A018121
KEYWORD
nonn
AUTHOR
Mircea Merca, Feb 20 2014
EXTENSIONS
Terms a(51) and beyond from Andrew Howroyd, May 01 2020
STATUS
approved