A028309 Molien series for ring of symmetrized weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 33, 38, 45, 51, 59, 66, 75, 83, 93, 102, 113, 123, 135, 146, 159, 171, 185, 198, 213, 227, 243, 258, 275, 291, 309, 326, 345, 363, 383, 402, 423, 443, 465, 486, 509, 531, 555, 578, 603, 627, 653, 678, 705, 731, 759, 786

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Programs

Magma

[n le 2 select Floor((n+2)/2) else (2*n^2-4*n+21+3*(-1)^n)/8: n in [0..50]]; // G. C. Greubel, Jan 05 2024

Mathematica

LinearRecurrence[{2,0,-2,1},{1,1,2,3,5,6,9},50] (* Harvey P. Dale, Nov 06 2016 *)

SageMath

[(2*n^2-4*n+21+3*(-1)^n)/8 - ((4-n)//2)*int(n<3) for n in range(51)] # G. C. Greubel, Jan 05 2024

Formula

G.f.: (1 - x + x^3 - x^5 + x^6)/((1-x)^2 * (1-x^2)). - Ralf Stephan, Apr 29 2014

a(n) = (1/8) * (2*n^2 + 3*(-1)^n - 4*n + 21) for n >= 3. - Ralf Stephan, Apr 29 2014 [Corrected by Pontus von Brömssen, May 30 2023]

From G. C. Greubel, Jan 05 2024: (Start)

a(n) = (1/8)*(2*n^2 - 4*n + 21 + 3*(-1)^n) - 2*[n=0] - [n=1] - [n=2].

E.g.f.: (1/8)*( (21 - 2*x + 2*x^2)*exp(x) + 3*exp(-x) ) - (2 + x + x^2/2). (End)