A238218 The total number of 3's in all partitions of n into an even number of distinct parts.

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

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

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).

a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Example

a(13) = 3 because the partitions in question are: 10+3, 7+3+2+1, 5+4+3+1.

Mathematica

nmax = 100; With[{k=3}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

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.

Cf. A067659, A067661.

Sequence in context: A309689 A029049 A094983 * A015744 A118301 A018121

Adjacent sequences: A238215 A238216 A238217 * A238219 A238220 A238221

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

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

Status

approved