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A004011
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Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.
(Formerly M5140)
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11
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1, 24, 24, 96, 24, 144, 96, 192, 24, 312, 144, 288, 96, 336, 192, 576, 24, 432, 312, 480, 144, 768, 288, 576, 96, 744, 336, 960, 192, 720, 576, 768, 24, 1152, 432, 1152, 312, 912, 480, 1344, 144, 1008, 768, 1056, 288, 1872, 576, 1152, 96, 1368, 744, 1728, 336
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| D_4 is also the Barnes-Wall lattice in 4 dimensions.
E_{gamma,2} is the unique normalized modular form for Gamma_0(2) of weight 2.
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REFERENCES
| H. Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. Eq. (1).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 214
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..10000
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
Michael Gilleland, Some Self-Similar Integer Sequences
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.
G. Nebe and N. J. A. Sloane, Home page for D_4 lattice
N. J. A. Sloane, The 24 minimal vectors form the 24-cell polytope
N. J. A. Sloane, Seven Staggering Sequences.
Eric Weisstein's World of Mathematics, 24-Cell
Eric Weisstein's World of Mathematics, Eisenstein Series
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Index entries for "core" sequences
Index entries for sequences related to D_4 lattice
Index entries for sequences related to Eisenstein series
Index entries for sequences related to Barnes-Wall lattices
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FORMULA
| a(0) = 1; if n>0 then a(n) = 24 (sum_{d|n, d odd, d>0} d) = 24 * A000593(n).
G.f.: 1 + 24 Sum_{n>0} n x^n /(1 + x^n). a(n) = A000118(2*n) = A096727(2*n).
G.f.: Sum_{ a, b, c, d} x^( a^2 + b^2 + c^2 + d^2 + a*d + b*d + c*d ). - Michael Somos, Jan 11 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 2*u*v - 7*v^2 - 8*v*w + 16*w^2. - Michael Somos, May 29 2005
Expansion of (1 + k^2) K(k^2)^2 / (pi/2)^2 in powers of nome q. - Michael Somos, Jun 10 2006
Expansion of b(x) * b(x^2) + 3 * c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions. - Michael Somos, Jan 11 2011
G.f.: (1/2) * (theta_3(z)^4 + theta_4(z)^4) = theta_3(2z)^4 + theta_2(2z)^4 = Sum_{k>=0} a(k) * x^(2*k).
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 2 (t/i)^2 f(t) where q = exp(2 pi i t). - Michael Somos, Sep 11 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 2*u1*u2 - 10*u1*u3 + 10*u1*u6 + 10*u2*u3 - 40*u2*u6 - 18*u3*u6. - Michael Somos, Sep 11 2007
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EXAMPLE
| 1 + 24*x + 24*x^2 + 96*x^3 + 24*x^4 + 144*x^5 + 96*x^6 + 192*x^7 + 24*x^8 + ...
1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + 192*q^14 + 24*q^16 + ...
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MAPLE
| readlib(ifactors): with(numtheory): for n from 1 to 100 do if n mod 2 = 0 then m := n/ifactors(n)[2][1][1]^ifactors(n)[2][1][2] else m := n fi: printf(`%d, `, 24*sigma(m)) od: # from James A. Sellers Dec 07 2000
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MATHEMATICA
| a[ n_] := If[ n < 0, 0, With[ {m = Floor @ Sqrt[4 n]}, SeriesCoefficient[ Sum[ q^( x^2 + y^2 + z^2 + t^2 + (x + y + z) t ), {x, -m, m}, {y, -m, m}, {z, -m, m}, {t, -m, m}] + O[q]^(n + 1), n]]] (* Michael Somos, Jan 11 2011 *)
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PROG
| (PARI) {a(n) = if( n<1, n==0, 24 * sumdiv( n, d, d%2 * d))} /* Michael Somos, Apr 17 2000 */
(PARI) {a(n) = if( n<1, n==0, qfrep([ 2, 1, 1, 1; 1, 2, 0, 0; 1, 0, 2, 0; 1, 0, 0, 2], n, 1)[n] * 2 )} /* Michael Somos, Sep 11 2007 */
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CROSSREFS
| Cf. A000118, A000593, A096727, A108092, A108096. Partial sums give A046949.
Cf. A108092 (fourth root).
Sequence in context: A022358 A122505 A103640 * A056465 A056455 A128378
Adjacent sequences: A004008 A004009 A004010 * A004012 A004013 A004014
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KEYWORD
| nonn,easy,core,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Barry Brent (barryb(AT)primenet.com)
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