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Hankel transform of central coefficients of (1+k*x-2x^2)^n, k arbitrary integer.
3

%I #13 Sep 08 2022 08:45:29

%S 1,-4,-32,512,16384,-1048576,-134217728,34359738368,17592186044416,

%T -18014398509481984,-36893488147419103232,151115727451828646838272,

%U 1237940039285380274899124224,-20282409603651670423947251286016

%N Hankel transform of central coefficients of (1+k*x-2x^2)^n, k arbitrary integer.

%C Hankel transform of A098332. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-2)x) and its k-th binomial transforms, are given by a(n). In general, the Hankel transform of e.g.f. Bessel_I(0,2*sqrt(m)x) and its binomial transforms is 2^n*m^C(n+1,2).

%C Unsigned version is A036442. - _Philippe Deléham_, Dec 11 2008

%H G. C. Greubel, <a href="/A127945/b127945.txt">Table of n, a(n) for n = 0..79</a>

%F a(n) = (cos(Pi*n/2) - sin(Pi*n/2))*4^n*2^C(n,2).

%F a(n) = 2^n*(-2)^C(n+1,2).

%t Table[2^n*(-2)^Binomial[n+1,2], {n, 0, 25}] (* _G. C. Greubel_, May 01 2018 *)

%o (PARI) for(n=0,25, print1(2^n*(-2)^binomial(n+1,2), ", ")) \\ _G. C. Greubel_, May 01 2018

%o (Magma) [2^n*(-2)^Binomial(n+1,2): n in [0..25]]; // _G. C. Greubel_, May 01 2018

%Y Cf. A036442, A098332.

%K easy,sign

%O 0,2

%A _Paul Barry_, Feb 08 2007