A238210 The total number of 3's in all partitions of n into an odd number of distinct parts.

0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 23, 28, 32, 37, 44, 51, 58, 68, 78, 89, 103, 118, 134, 154, 175, 199, 227, 257, 291, 330, 373, 421, 475, 535, 602, 677, 760, 852, 955, 1069, 1196, 1336, 1491, 1663, 1853, 2063, 2296

Offset

0,10

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/6)} A067661(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067659(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(12) = 3 because the partitions in question are: 8+3+1, 7+3+2, 5+4+3.

Prog

(PARI) seq(n)={my(A=O(x^(n-2))); Vec(x^3*(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 A238450.

Cf. A067659, A067661.

Sequence in context: A035397 A033295 A018049 * A120170 A136421 A274159

Adjacent sequences: A238207 A238208 A238209 * A238211 A238212 A238213

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

Status

approved