0,2

Also the number of jagged partitions of n.

Euler transform of period 2 sequence [2,1,...].

According to Ramanujan (1913) a(n) is close to (cosh(x)-sinh(x)/x)/(4n) where x=pi*sqrt(n).

Number of partitions of 2n with all odd parts occurring with even multiplicities. There is no restriction on the even parts. Cf. A006950, A046682. - Mamuka Jibladze (jib(AT)rmi.acnet.ge), Sep 05 2003

Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras sp(n), n=0,1,2,3,... (the case n=0 being degenerate). A006950, this sequence and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003

Also, number of 01-partitions of n. A 01-partition of n is a weakly decreasing sequence of m nonnegative integers n(i) such that sum(n(i))=n, n(m)>0, n(j)>=n(j+1)-1 and n(j)>=n(j+2). They are special cases of jagged partitions.

a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).

Smallest sequence of even numbers (except a(0)) which is the Euler transform of a sequence of positive integers. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 16 2006

Equals A022567 convolved with A035363 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09 2009]

Equals the infinite product [1,2,2,2,...] * [1,0,2,0,2,0,2,...] * [1,0,0,2,0,0,2,0,0,2,...] * ... [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 05 2009]

G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.

S. Corteel, Particle seas and basic hypergeometric series, Advances Appl. Math., 31 (2003), 199-214.

S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ranaujan's {}_1 psi_1 summation, J. Combin. Theory A 97 (2002), 177-183.

S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc., 356 (2004), 1623-1635.

K. Mahlburg, The overpartition function modulo small powers of 2, Discr. Math., 286 (2004), 263-267.

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.

J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory A 103 (2003), 393-401.

J. R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.

I. Pak, Partition bijections, a survey, Ramanujan J., to appear.

T. D. Noe, Table of n, a(n) for n=0..1000

G. Almkvist, Asymptotic formulas and generalized Dedekind sums.

N. Chair, Partition identities from Partial Supersymmetry

J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions

G.f.: Product_{m=1..inf} (1+q^m)/(1-q^m); also (Sum (-q)^(m^2), m = -inf .. inf )^(-1).

Taylor series for 1/theta_4.

Product expansion is 1 / product_{m=1..inf} (1-q^(2m)) * ( 1-q^(2m-1))^2.

Convolution of A000041 and A000009. - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 26 2002

Recurrence: a(n) = 2*Sum[m>=1, (-1)^(m+1) * a(n-m^2)].

a(n) = (1/n)*Sum_{k=1..n} (sigma(2*k)-sigma(k))*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 05 2004

G.f. : product(i=1, oo, (1+x^i)^A001511(2i)) (see A000041) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004

Expansion of eta(q)^2 / eta(q^2) in powers of q. - - Michael Somos Nov 01 2008

Expansion of 1 / phi(-q) in powers of q where phi() is a Ramanujan theta function. - - Michael Somos Nov 01 2008

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^4 * (u^4 + v^4) - 2 * u^2 * v^6. - Michael Somos Nov 01 2008

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6^3 * (u1^2 + u3^2) - 2 * u1 * u2 * u3^3. - Michael Somos Nov 01 2008

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2^3 * (u3^2 - 3 * u1^2) + 2 * u1^3 * u3 * u6. - Michael Somos Nov 01 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(-1/2) (t / i)^(-1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A106507. - Michael Somos Nov 01 2008

G.f.: exp( Sum_{n>=1} 2*x^(2n-1)/(1 - x^(2n-1))/(2n-1) ). [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 06 2009]

1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 40*q^6 + 64*q^7 + 100*q^8 + ...

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A)^2, n))} /* Michael Somos Nov 01 2008 */

(PARI) {a(n)=polcoeff(exp(sum(m=1, n\2+1, 2*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))/(2*m-1))), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 06 2009]

Convolution inverse of A002448. A004402(n) = (-1)^n * a(n).

A022567, A035363 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09 2009]

Sequence in context: A069252 A069253 A004402 this_sequence A123655 A084683 A118544

Adjacent sequences: A015125 A015126 A015127 this_sequence A015129 A015130 A015131

nonn,easy,nice

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