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A008438
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Sum of divisors of 2*n + 1.
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74
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1, 4, 6, 8, 13, 12, 14, 24, 18, 20, 32, 24, 31, 40, 30, 32, 48, 48, 38, 56, 42, 44, 78, 48, 57, 72, 54, 72, 80, 60, 62, 104, 84, 68, 96, 72, 74, 124, 96, 80, 121, 84, 108, 120, 90, 112, 128, 120, 98, 156, 102, 104, 192, 108, 110, 152, 114, 144, 182, 144, 133, 168
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OFFSET
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0,2
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COMMENTS
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Number of ways of writing n as the sum of 4 triangular numbers.
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
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139 Ex. (iii).
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 eq. (6), and p. 283 eq. (8).
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).
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.
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LINKS
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FORMULA
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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
Expansion of Jacobi theta_2(q)^4 / (16*q) 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
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
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
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).
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
G.f.: Sum_{k>=0} (2k + 1) * x^k / (1 - x^(2k + 1)).
G.f.: (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Apr 11 2004
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
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
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
G.f.: Sum_{n >= 0} x^n*(1 + x^(2*n+1))/(1 - x^(2*n+1))^2. (End)
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EXAMPLE
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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 + ...
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 + ...
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 + ...
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MAPLE
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MATHEMATICA
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DivisorSigma[1, 2 # + 1] & /@ Range[0, 61] (* Ant King, Dec 02 2010 *)
a[ n_] := SeriesCoefficient[ D[ Series[ Log[ QPochhammer[ -x] / QPochhammer[ x]], {x, 0, 2 n + 1}], x], {x, 0 , 2n}]; (* Michael Somos, Oct 15 2019 *)
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PROG
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(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) = 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 */
(Sage) ModularForms( Gamma0(4), 2, prec=124).1; # Michael Somos, Jun 12 2014
(Magma) Basis( ModularForms( Gamma0(4), 2), 124) [2]; /* Michael Somos, Jun 12 2014 */
(Haskell)
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CROSSREFS
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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.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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