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%I #132 Nov 03 2023 10:44:10
%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 2*n + 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
%C a(n) is also the total number of parts in all partitions of 2*n + 1 into equal parts. - _Omar E. Pol_, Feb 14 2021
%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139 Ex. (iii).
%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 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 184, Prop. 4, F(z).
%D G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ. Press, 1954, page 92 ff.
%H N. J. A. Sloane, <a href="/A008438/b008438.txt">Table of n, a(n) for n = 0..20000</a> (first 10000 terms from T. D. Noe)
%H H. Cohen, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002311860">Sums involving the values at negative integers of L-functions of quadratic characters</a>, Math. Ann. 217 (1975), no. 3, 271-285. MR0382192 (52 #3080)
%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211
%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
%H K. Ono, S. Robins and P. T. Wahl, <a href="http://dx.doi.org/10.1007/BF01831114">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Theorem 3 [Legendre].
%H H. Rosengren, <a href="http://arXiv.org/abs/math.NT/0504272">Sums of triangular numbers from the Frobenius determinant</a>, arXiv:math/0504272 [math.NT], 2005.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Min Wang, Zhi-Hong Sun, <a href="http://arxiv.org/abs/1511.00478">On the number of representations of n as a linear combination of four triangular numbers II</a>, arXiv:1511.00478 [math.NT], 2015.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H K. S. Williams, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.04.329">The parents of Jacobi's four squares theorem are unique</a>, Amer. Math. Monthly, 120 (2013), 329-345.
%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() 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(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^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(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^3), B(q^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(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^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 - 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(2*n+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
%F a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - _Seiichi Manyama_, May 06 2017
%F G.f.: exp(Sum_{k>=1} 4*(x^k/k)/(1 + x^k)). - _Ilya Gutkovskiy_, Jul 31 2017
%F From _Peter Bala_, Jan 10 2021: (Start)
%F a(n) = A002131(2*n+1).
%F G.f.: Sum_{n >= 0} x^n*(1 + x^(2*n+1))/(1 - x^(2*n+1))^2. (End)
%F Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 8. - _Vaclav Kotesovec_, Aug 07 2022
%F Convolution of A125061 and A138741. - _Michael Somos_, Mar 04 2023
%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 B(q) = 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 *)
%t a[ n_] := SeriesCoefficient[ D[ Series[ Log[ QPochhammer[ -x] / QPochhammer[ x]], {x, 0, 2 n + 1}], x], {x, 0 , 2n}]; (* _Michael Somos_, Oct 15 2019 *)
%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) = my(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 */
%o (Haskell)
%o a008438 = a000203 . a005408 -- _Reinhard Zumkeller_, Sep 22 2014
%o (Magma) [DivisorSigma(1, 2*n+1): n in [0..70]]; // _Vincenzo Librandi_, Aug 01 2017
%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.
%Y Cf. A000203, A002131, A005408, A062731, A099774, A125061, A138741.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Comments from _Len Smiley_, _Enoch Haga_
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