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A247984
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Constant terms of congruences derived from a permutation polynomial motivated by generalized quadrangles.
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2
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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
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = 2^n when n is odd and a(n) = 2^n - (-1)^(n/2)*binomial(n, n/2) when n is even.
G.f.: 1/(1-2*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*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
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MAPLE
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if type(n, 'odd') then
2^n;
else
2^n-(-1)^(n/2)*binomial(n, n/2) ;
end if;
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = if (n % 2, 2^n, 2^n - (-1)^(n/2)*binomial(n, n/2)); \\ Michel Marcus, Oct 01 2014
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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