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A008438 Sum of divisors of 2n + 1. 39

%I

%S 1,4,6,8,13,12,14,24,18,20,32,24,31,40,30,32,48,48,38,56,42,44,78,48,

%T 57,72,54,72,80,60,62,104,84,68,96,72,74,124,96,80,121,84,108,120,90,

%U 112,128,120,98,156,102,104,192,108,110,152,114,144,182,144,133,168

%N Sum of divisors of 2n + 1.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

%C Bisection of A000203. - _Omar E. Pol_, Mar 14 2012

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139 Ex. (iii).

%D 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)

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

%D 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 eq. (6), and p. 283 eq. (8).

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

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

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

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

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

%D M. Koike, Modular forms on non-compact arithmetic triangle groups, preprint.

%D K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94; http://www.math.wisc.edu/~ono/reprints/006.pdf.

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

%D K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.

%H T. D. Noe, <a href="/A008438/b008438.txt">Table of n, a(n) for n=0..10000</a>

%H H. Rosengren, <a href="http://arXiv.org/abs/math.NT/0504272">Sums of triangular numbers from the Frobenius determinant</a>

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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

%F Expansion of Jacobi theta_2(q)^4 / (16*q) in powers of q^2. - _Michael Somos_, Apr 11 2004

%F Euler transform of period 2 sequence [ 4, -4, 4, -4, ...]. - _Michael Somos_, Apr 11 2004

%F a(n) = b(2*n + 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

%F 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

%F 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).

%F 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

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

%F G.f.: (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - _Michael Somos_, Apr 11 2004

%F G.f. Sum_{k>=0} a(k) * x^(2k + 1) = x( * Prod_{k>0} (1 - x^(4*k))^2 / (1 - x^(2k)))^ 4 = x * (Sum_{k>0} x^(k^2 - k))^4 = Sum_{k>0} k * (x^k / (1 - x^k) - 3 * x^(2*k) / (1 - x^(2*k)) +2 * x^(4*k) / (1 - x^(4*k))). - _Michael Somos_, Jul 07 2004

%F Number of solutions of 2*n + 1 = (x^2 + y^2 + z^2 + w^2) / 4 in positive odd integers. - _Michael Somos_, Apr 11 2004

%F 8 * a(n) = A005879(n) = A000118(2*n + 1). 16 * a(n) = A129588(n). a(n) = A000593(2*n + 1) = A115607(2*n + 1).

%F a(n) = A000203(2n+1). - _Omar E. Pol_, Mar 14 2012

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A096727. _Michael Somos_, Jun 12 2014

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

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

%e G.f. = 1 + 4*x + 6*x^2 + 8*x^3 + 13*x^4 + 12*x^5 + 14*x^6 + 24*x^7 + 18*x^8 + 20*x^9 + ...

%e G.f. = q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + 18*q^17 + ...

%p A008438 := proc(n) numtheory[sigma](2*n+1) ; end proc: # _R. J. Mathar_, Mar 23 2011

%t DivisorSigma[1, 2 # + 1] & /@ Range[0, 61] (* _Ant King_, Dec 02 2010 *)

%o (PARI) {a(n) = if( n<0, 0, sigma( 2*n + 1))};

%o (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 */

%o (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 */

%o (Sage) ModularForms( Gamma0(4), 2, prec=124).1; # _Michael Somos_, Jun 12 2014

%o (MAGMA) Basis( ModularForms( Gamma0(4), 2), 124) [2]; /* _Michael Somos_, Jun 12 2014 */

%Y Cf. A000118, A000593, A005879, A096727, A115607, A129588, A225699/A225700.

%Y Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_.

%E Comments from Len Smiley (smiley(AT)math.uaa.alaska.edu), _Enoch Haga_

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