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

0,11

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/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).

Example

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

Mathematica

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

Prog

(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

Crossrefs

Column k=2 of A238451.

Cf. A067659, A067661.

Sequence in context: A265771 A239513 A029018 * A185226 A096765 A025147

Adjacent sequences: A238214 A238215 A238216 * A238218 A238219 A238220

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

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

Status

approved