0,2

Number of partitions of n into distinct parts, with 2 types of each part. E.g. for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*, thus a(4)=9 - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004

Number of partitions of n into odd parts, each part being of two kinds. E.g. a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2005

Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2005

Equals A000041 convolved with A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 852

a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 09 2004

Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q.

Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function.

G.f. is a period 1 Fourier series which satisfies f(-1/(288 t)) = (1/2) / f(t) where q = exp(2 pi i t).

Parity result: a(n) is even except when n is twice a generalised pentagonal number (i.e. of the form 2*A001318(m) for some m). [From Peter Bala (pbala(AT)talktalk.net), Mar 19 2009]

q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...

(PARI) a(n)=if(n<0, 0, polcoeff(prod(k=1, n, 1+x^k, 1+x*O(x^n))^2, n))

(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A))^2, n))

Convolution square of A000009. Convolution inverse of A022597.

A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]

Sequence in context: A094056 A058609 A128518 this_sequence A134004 A123631 A018060

Adjacent sequences: A022564 A022565 A022566 this_sequence A022568 A022569 A022570

nonn

N. J. A. Sloane (njas(AT)research.att.com).