A238211 - OEIS

A238211

The total number of 4's in all partitions of n into an odd number of distinct parts.

0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 5, 5, 6, 7, 9, 11, 13, 15, 18, 21, 25, 29, 34, 40, 46, 54, 62, 71, 82, 95, 108, 124, 142, 162, 184, 210, 238, 271, 306, 346, 392, 443, 498, 561, 632, 710, 796, 893, 1000, 1120, 1252, 1397, 1560, 1740, 1937, 2156

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OFFSET

0,12

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = Sum_{j=1..round(n/8)} A067661(n-(2*j-1)*4) - Sum_{j=1..floor(n/8)} A067659(n-8*j).

G.f.: (1/2)*(x^4/(1+x^4))*(Product{n>=1} 1 + x^n) + (1/2)*(x^4/(1-x^4))*(Product_{n>=1} 1 - x^n).

EXAMPLE

a(12) = 3 because the partitions in question are: 7+4+1, 6+4+2, 5+4+3.