0,2

Number of ways of writing n as the sum of 4 triangular numbers.

H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285. MR0382192 (52 #3080)

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

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 19.

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 12.

H. M. Farkas, I. Kra, Cosines and triangular numbers, Rev. Roumaine Math. Pures Appl., 46 (2001), 37-43.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.31).

M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 184, Prop. 4, F(z)..

G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ. Press, 1954, page 92 ff.

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

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

Expansion of q^(-1/2)(eta(q^2)^2/eta(q))^4 = psi(q)^4 in powers of q where psi is a Ramanujan theta function. - Michael Somos Apr 11 2004

G.f.: (Product_{k>0} (1-x^k)(1+x^k)^2)^4. - Michael Somos Apr 11 2004

Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^3+8*w*v^2 + 16*w^2*v - u^2*w - Michael Somos Apr 08 2005

G.f. Sum_{k>=0} a(k)x^(2k+1) = x(Prod_{k>0} (1-x^(4k))^2/(1-x^(2k)))^ 4 = x(Sum_{k>0} x^(k^2-k))^4 = Sum_{k>0} k(x^k/(1-x^k) -3x^(2k)/(1-x^(2k)) +2x^ (4k)/(1-x^(4k))). - Michael Somos Jul 07 2004

Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6) where f(u1, u2, u3, u6)=u2^3+u1^2*u6+3*u2*u3^2+27*u6^3-u1*u2*u3-3*u1*u3*u6-7*u2^2*u6-21*u2*u6^2 . - Michael Somos May 30 2005

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

a(n)=b(2n+1) where b(n) is multiplicative and b(2^e)=0^n, b(p^e)=(p^(e+1)-1)/(p-1) if p>2. - Michael Somos Jul 07 2004

Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^3), B(x^9)) where f(u, v, w) = v^4 -30*u*v^2*w +12*u*v*w*(u +9*w) -u*w*(u^2 +9*w*u +81*w^2).

Expansion of Jacobi theta_2(q)^4/(16q) in powers of q^2. - Michael Somos Apr 11 2004

Euler transform of period 2 sequence [4,-4,4,-4,...]. - Michael Somos Apr 11 2004

Number of solutions of 2n+1 = (x^2+y^2+z^2+w^2)/4 in positive odd integers. - Michael Somos Apr 11 2004

Divisors of 9 are 1,3,9, so a(4)=1+3+9=13.

F_2(z) = eta(4z)^8/eta(2z)^4 = q + 4q^3 + 6q^5 +8q^7 + 13q^9 + ...

lst={}; Do[AppendTo[lst, DivisorSigma[1, n]], {n, 1, 6!, 2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 20 2008]

(PARI) a(n)=if(n<0, 0, sigma(2*n+1))

(PARI) a(n)= if(n<0, 0, n=2*n; polcoeff(sum(k=1, (sqrtint(4*n+1)+1)\2, x^(k^2-k), x*O(x^n))^4, n)) /* Michael Somos Sep 17 2004 */

(PARI) {a(n)= local(A); if(n<0, 0, n=2*n; A=x*O(x^n); polcoeff( (eta(x^4+A))^2/eta(x^2+A))^4, n))} /* Michael Somos Sep 17 2004 */

A005879(n)=8a(n). A129588(n)=16a(n).

Sequence in context: A146528 A020153 A151760 this_sequence A121613 A145284 A023560

Adjacent sequences: A008435 A008436 A008437 this_sequence A008439 A008440 A008441

nonn,easy,nice

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

Comments from Len Smiley (smiley(AT)math.uaa.alaska.edu), Enoch Haga (Enokh(AT)comcast.net)