0,2

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139 Example (iv).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 72, Eq (31.2); p. 78, Eq. following (32.25).

Gosper, R. W. "Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics.", in Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Dekker, 1990. p. 279.

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 31, Article 272.

T. D. Noe, Table of n, a(n) for n=0..10000

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

R. W. Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics.

H. Rosengren, Sums of triangular numbers from the Frobenius determinant

Fine gives an explicit formula for a(n) in terms of the divisors of n.

a(n) = the difference between the number of divisors of 4*n+1 of form 4*k+1 and the number of form 4*k-1. - David Broadhurst, Oct 20 2002

G.f.: (Sum_{k>=0} x^((k^2+k)/2))^2 = (Sum_{k>=0} x^(k^2+k))(Sum_k x^(k^2)).

Expansion of Jacobi theta constant (theta_2(z/2))^2/(4q^(1/4)).

G.f. = s(2)^4/(s(1)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

Sum[d|(4n+1), (-1)^((d-1)/2) ].

a(n)=b(4n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos Sep 14 2005

Given g.f. A(x), then B(x)=xA(x^4) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^3+4*v*w^2-u^2*w . - Michael Somos Sep 14 2005

Given g.f. A(x), then B(x)=xA(x^4) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1*u3-(u2-u6)*(u2+3*u6) . - Michael Somos Sep 14 2005

Expansion of Jacobi q^(-1/2) kK/(2pi) in powers of q^2. - Michael Somos Sep 14 2005

G.f.: Sum_{k>=0} a(k)x^(2k) = Sum_{k>=0} x^k/(1+x^(2k+1)).

G.f.: Sum_{k} x^k/(1-x^(4k+1)) . - Michael Somos Nov 03 2005

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

Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos Jan 25 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/2 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A104794.

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

Expansion of q^(-1/4) * eta(q^2)^4 / eta(q)^2 in powers of q.

q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

(PARI) a(n)=if(n<1, n==0, polcoeff(sum(k=0, (sqrtint(8*n+1)-1)\2, x^(k*(k+1)/2), x*O(x^n))^2, n))

(PARI) a(n)=if(n<0, 0, n=4*n+1; sumdiv(n, d, (-1)^(d\2))) /* Michael Somos Sep 02 2005 */

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

(PARI) a(n)=if(n<0, 0, n=4*n+1; sumdiv(n, d, (d%4==1)-(d%4==3))) /* Michael Somos Sep 14 2005 */

A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113414, A113446, A113652 all satisfy A(4n+1)=a(n).

A004020(n)=2*a(n). A005883(n)=4*a(n).

Adjacent sequences: A008438 A008439 A008440 this_sequence A008442 A008443 A008444

Sequence in context: A058496 A137579 A108805 this_sequence A134343 A108804 A127249

nonn,easy

njas

More terms and information from Michael Somos, Mar 23 2003