A028344 - OEIS

%I #28 Mar 03 2024 10:57:52

%S 1,1,1,3,4,9,19,27,41,62,91,128,175,231,298,392,498,617,768,933,1134,

%T 1372,1627,1922,2260,2642,3070,3546,4071,4647,5307,6021,6792,7654,

%U 8576,9595,10714,11898,13187,14584

%N Molien series for complete weight enumerator of self-dual code over GF(5).

%D Gagola, Stephen M., Jr. Weight enumerators of normalized codes. SIAM J. Algebraic Discrete Methods 2 (1981), no. 4, 347-380. MR0634360 (83e:94046). See Appendix, unnormalized codes, case q=5. _N. J. A. Sloane_, Apr 09 2014

%H G. C. Greubel, <a href="/A028344/b028344.txt">Table of n, a(n) for n = 0..1000</a>

%H E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (<a href="http://neilsloane.com/doc/self.txt">Abstract</a>, <a href="http://neilsloane.com/doc/self.pdf">pdf</a>, <a href="http://neilsloane.com/doc/self.ps">ps</a>).

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 2, 0, 0, -1, -2, -3, 0, 3, 2, 1, 0, 0, -2, -1, 0, 1).

%F G.f.: (x^13 + 3*x^12 + 5*x^11 + 6*x^10 + 5*x^9 + 10*x^8 + 13*x^7 + 10*x^6 + 4*x^5 + x^4 + x + 1)/(-x^18 + x^16 + 2*x^15 - x^12 - 2*x^11 - 3*x^10 + 3*x^8 + 2*x^7 + x^6 - 2*x^3 - x^2 + 1) = (x^13 + 3*x^12 + 5*x^11 + 6*x^10 + 5*x^9 + 10*x^8 + 13*x^7 + 10*x^6 + 4*x^5 + x^4 + x + 1)/((1-x)^5*(1+x)*(1+x+x^2)^2*(1+x+x^2+x^3+x^4)^2). - _Charles R Greathouse IV_, Dec 08 2016

%p # Another program, from _N. J. A. Sloane_, Apr 09 2014

%p a:=[1,1,0,0,1,4,10,13,10,5,6,5,3,1];

%p t:=n->1-x^n;

%p P:=add(a[i]*x^(i-1),i=1..nops(a))/(t(2)*t(3)^2*t(5)^2);

%p f:=tt->seriestolist(series(tt,x,80));

%t Take[CoefficientList[Series[(t^26 + 3*t^24 + 5*t^22 + 6*t^20 + 5*t^18 + 10*t^16 + 13*t^14 + 10*t^12 + 4*t^10 + t^8 + t^2 + 1)/( (1 - t^4)*(1 - t^6)^2*(1 - t^10)^2 ), {t, 0, 40}], t], {1, -1, 2}] (* _G. C. Greubel_, Nov 24 2016 *)

%t LinearRecurrence[{0, 1, 2, 0, 0, -1, -2, -3, 0, 3, 2, 1, 0, 0, -2, -1, 0, 1}, {1, 1, 1, 3, 4, 9, 19, 27, 41, 62, 91, 128, 175, 231, 298, 392, 498, 617}, 50] (* _Robert G. Wilson v_, Nov 25 2016 *)

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_