0,2

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

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

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....

G.f.: eta(q^4)^2/(eta(q)^2*q^(1/4)) where eta = Dedekind's function.

A079006(n) = (-1)^n a(n).

Expansion of q^(-1/4)(1/2)(k/k')^(1/2) in powers of q.

Euler transform of period 4 sequence [2, 2, 2, 0, ...].

Given g.f. A(x), then B(x)=(x*A(x^4))^4 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(1+16u)(1+16v)v-u^2 . - Michael Somos Jul 09 2005

Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^2+v^2)^2 -uv(1+4uv)^2 . - Michael Somos Jul 09 2005

G.f.: (Product_{k>0} (1+x^(2k))/(1-x^(2k-1)))^2 = (Product_{k>0} (1-x^(4k))/(1-x^k))^2 = square of g.f. for A001935.

with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr (n-> [2, 2, 2, 0] [modp(n-1, 4)+1]): seq (a(n), n=0..38); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 08 2008]

(PARI) a(n)=if(n<0, 0, polcoeff((eta(x^4+x*O(x^n))/eta(x+x*O(x^n)))^2, n))

(PARI) a(n)=if(n<0, 0, polcoeff(prod(k=1, n, 1/if(k%4, 1-x^k, 1), 1+x*O(x^n))^2, n))

See A127391, A127392, A079006 for other versions of this sequence.

A079006(n)=(-1)^n a(n). Convolution square of A001935.

Adjacent sequences: A001933 A001934 A001935 this_sequence A001937 A001938 A001939

Sequence in context: A006327 A103577 A079006 this_sequence A127297 A018739 A011893

nonn

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