A238217 The total number of 2's in all partitions of n into an even number of distinct parts.

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

Author

Mircea Merca, Feb 20 2014

Keywords

• nonn

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).

Examples

			a(12) = 3 because the partitions in question are: 10+2, 6+3+2+1, 5+4+2+1.
		

Links

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

Crossrefs

Column k=2 of A238451.

Cf. A067659, A067661.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Length[#] == Length[ Union[#]]&&MemberQ[#,2]&]],{n,0,50}] (* Harvey P. Dale, Dec 09 2014 *)
nmax = 100; With[{k=2}, 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 *)

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

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).

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

Extensions

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