

A247984


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


2



2, 6, 8, 10, 32, 84, 128, 186, 512, 1276, 2048, 3172, 8192, 19816, 32768, 52666, 131072, 310764, 524288, 863820, 2097152, 4899736, 8388608, 14073060, 33554432, 77509464, 134217728, 228318856, 536870912, 1228859344, 2147483648
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OFFSET

1,1


COMMENTS

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 HermiteDickson 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.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Dmytrenko, F. Lazebnik, and J. Williford, On monomial graphs of girth eight, Finite Fields and Their Applications 13 (2007), 828842.
Brian G. Kronenthal, Monomial Graphs and Generalized Quadrangles, Finite Fields and Their Applications, 18 (2012), 674684.
B. G. Kronenthal, An Integer Sequence Motivated by Generalized Quadrangles, Journal of Integer Sequences, 2015, Vol. 18. #15.7.8.


FORMULA

a(n) = 2^n when n is odd and a(n) = 2^n  (1)^(n/2)*binomial(n, n/2) when n is even.
From Robert Israel, Oct 01 2014: (Start)
G.f.: 1/(12*x)  1/sqrt(1+4*x^2).
E.g.f.: exp(2*x)  J_0(2*x) where J_0 is a Bessel function. (End)
n*(2*n3)*a(n) 2*(2*n1)*(n1)*a(n1) +4*(n1)*(2*n3)*a(n2) 8*(2*n1)*(n2)*a(n3)=0.  R. J. Mathar, Jun 09 2018


MAPLE

A247984 := proc(n)
if type(n, 'odd') then
2^n;
else
2^n(1)^(n/2)*binomial(n, n/2) ;
end if;
end proc: # R. J. Mathar, Jun 09 2018


MATHEMATICA

For[n = 1, n < 31, n++, Piecewise[{{Print[2^n  (1)^(n/2) * Binomial[n, n/2]], EvenQ[n]}, {Print[2^n], OddQ[n]}}]]
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 *)


PROG

(PARI) a(n) = if (n % 2, 2^n, 2^n  (1)^(n/2)*binomial(n, n/2)); \\ Michel Marcus, Oct 01 2014


CROSSREFS

Cf. A246800.
Sequence in context: A066762 A066198 A299381 * A108417 A185078 A214402
Adjacent sequences: A247981 A247982 A247983 * A247985 A247986 A247987


KEYWORD

easy,nonn


AUTHOR

Brian G. Kronenthal, Sep 28 2014


STATUS

approved



