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URL: https://oeis.org/A028887

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A028887
Theta series of 4-dimensional 5-modular lattice with det 25 and minimal norm 2.
5
1, 6, 18, 24, 42, 6, 72, 48, 90, 78, 18, 72, 168, 84, 144, 24, 186, 108, 234, 120, 42, 192, 216, 144, 360, 6, 252, 240, 336, 180, 72, 192, 378, 288, 324, 48, 546, 228, 360, 336, 90, 252, 576, 264, 504, 78, 432, 288, 744, 342, 18, 432, 588, 324, 720, 72, 720, 480
OFFSET
0,2
REFERENCES
Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 463, Entry 4(i).
LINKS
Shaun Cooper and Dongxi Ye, Level 14 and 15 Analogues of Ramanujan's Elliptic Functions to Alternative Bases, Trans. Amer. Math. Soc., Vol. 368, No. 11 (2016), pp. 7883-7910; ResearchGate preprint, 2015.
Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.
FORMULA
a(n) = 6*b(n) where b(n) is multiplicative with a(0) = 1, b(5^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise. - Michael Somos, Feb 04 2006
G.f. 1 + 6 * (Sum_{k>0} k * x^k / (1 - x^k) - 5*k * x^(5*k) / (1 - x^(5*k))). - Michael Somos, Feb 04 2006
Sum_{k=0..n} a(k) ~ (2*Pi^2/5) * n^2. - Amiram Eldar, Nov 24 2025
EXAMPLE
G.f. = 1 + 6*x + 18*x^2 + 24*x^3 + 42*x^4 + 6*x^5 + 72*x^6 + 48*x^7 + ...
G.f. = 1 + 6*q^2 + 18*q^4 + 24*q^6 + 42*q^8 + 6*q^10 + 72*q^12 + 48*q^14 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[ n == 0], 6 Sum[ If[ Mod[ d, 5] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ 1 + 6 Sum[ k x^k / (1 - x^k) - 5 k x^(5 k) / (1 - x^(5 k)), {k, n}], {x, 0, n}]; (* Michael Somos, Jun 12 2014 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 6 * sumdiv(n, d, (d%5>0) * d))}; /* Michael Somos, Feb 04 2006 */
(PARI) {a(n) = my(G); if( n<0, 0, G = [ 2, 1, 0, 0; 1, 2, 1, 0; 0, 1, 4, 5; 0, 0, 5, 10]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n, 1)), n))}; /* Michael Somos, Jun 12 2014 */
(SageMath) ModularForms( Gamma0(5), 2, prec=70).0; # Michael Somos, Jun 12 2014
(Magma) Basis( ModularForms( Gamma0(5), 2), 70) [1]; /* Michael Somos, Jun 12 2014 */
CROSSREFS
Sequence in context: A236864 A372566 A101527 * A283118 A274536 A051395
KEYWORD
nonn,easy
STATUS
approved