cp's OEIS Frontend

A247984 Constant terms of congruences derived from a permutation polynomial motivated by generalized quadrangles.

%I A247984 #28 Jun 09 2018 08:33:46
%S A247984 2,6,8,10,32,84,128,186,512,1276,2048,3172,8192,19816,32768,52666,
%T A247984 131072,310764,524288,863820,2097152,4899736,8388608,14073060,
%U A247984 33554432,77509464,134217728,228318856,536870912,1228859344,2147483648
%N A247984 Constant terms of congruences derived from a permutation polynomial motivated by generalized quadrangles.
%C A247984 Let p be an odd prime, let e be a positive integer, and let q = p^e. Dmytrenko, Lazebnik, and Williford (2007) proved that every monomial graph of girth at least eight is isomorphic to G = G_q(xy, x^ky^(2k)) for some integer k which is not divisible by p. If q = 3, then G is isomorphic to G_3(xy, xy^2). If q >= 5, then F(x) = ((x + 1)^(2k) - 1)x^(q - 1 - k) - 2x^(q - 1) is a permutation polynomial, in which case the Hermite-Dickson Criterion implies that the coefficient at x^(q - 1) in F(x)^n must equal 0 modulo p. a(n) is the constant term of the coefficient at x^(q - 1) in F(x)^n; this was first stated in Kronenthal (2012). The provided Mathematica program produces the first 30 terms of the sequence.
%H A247984 G. C. Greubel, <a href="/A247984/b247984.txt">Table of n, a(n) for n = 1..1000</a>
%H A247984 V. Dmytrenko, F. Lazebnik, and J. Williford, <a href="http://dx.doi.org/10.1016/j.ffa.2006.03.001">On monomial graphs of girth eight</a>, Finite Fields and Their Applications 13 (2007), 828-842.
%H A247984 Brian G. Kronenthal, <a href="http://dx.doi.org/10.1016/j.ffa.2012.01.001">Monomial Graphs and Generalized Quadrangles</a>, Finite Fields and Their Applications, 18 (2012), 674-684.
%H A247984 B. G. Kronenthal, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kronenthal/kron3.html">An Integer Sequence Motivated by Generalized Quadrangles</a>, Journal of Integer Sequences, 2015, Vol. 18. #15.7.8.
%F A247984 a(n) = 2^n when n is odd and a(n) = 2^n - (-1)^(n/2)*binomial(n, n/2) when n is even.
%F A247984 From _Robert Israel_, Oct 01 2014: (Start)
%F A247984 G.f.: 1/(1-2*x) - 1/sqrt(1+4*x^2).
%F A247984 E.g.f.: exp(2*x) - J_0(2*x) where J_0 is a Bessel function. (End)
%F A247984 n*(2*n-3)*a(n) -2*(2*n-1)*(n-1)*a(n-1) +4*(n-1)*(2*n-3)*a(n-2) -8*(2*n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Jun 09 2018
%p A247984 A247984 := proc(n)
%p A247984     if type(n,'odd') then
%p A247984         2^n;
%p A247984     else
%p A247984         2^n-(-1)^(n/2)*binomial(n, n/2) ;
%p A247984     end if;
%p A247984 end proc: # _R. J. Mathar_, Jun 09 2018
%t A247984 For[n = 1, n < 31, n++, Piecewise[{{Print[2^n - (-1)^(n/2) * Binomial[n, n/2]], EvenQ[n]}, {Print[2^n], OddQ[n]}}]]
%t A247984 Rest[With[{nn = 50}, CoefficientList[Series[Exp[2*x] - BesselJ[0, 2*x], {x, 0, nn}], x]*Range[0, nn]!]] (* _G. C. Greubel_, Aug 16 2017 *)
%o A247984 (PARI) a(n) = if (n % 2, 2^n, 2^n - (-1)^(n/2)*binomial(n, n/2)); \\ _Michel Marcus_, Oct 01 2014
%Y A247984 Cf. A246800.
%K A247984 easy,nonn
%O A247984 1,1
%A A247984 _Brian G. Kronenthal_, Sep 28 2014