A238209 The total number of 2's in all partitions of n into an odd number of distinct parts.

0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 13, 16, 18, 22, 26, 30, 35, 41, 48, 55, 64, 73, 85, 97, 111, 127, 146, 165, 189, 214, 244, 276, 313, 353, 400, 451, 508, 572, 644, 722, 811, 909, 1018, 1139, 1273, 1421, 1586, 1768, 1968, 2191, 2436

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/4)} A067661(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067659(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(11) = 3 because the partitions in question are: 8+2+1, 6+3+2, 5+4+2.

Prog

(PARI) seq(n)={my(A=O(x^(n-1))); Vec(x^2*(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 A238450.

Cf. A067659, A067661.

Sequence in context: A025149 A026799 A185326 * A342096 A210716 A027190

Adjacent sequences: A238206 A238207 A238208 * A238210 A238211 A238212

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

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

Status

approved