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%I #30 Mar 13 2023 16:43:23
%S 1,-3,2,-3,6,-14,36,-99,286,-858,2652,-8398,27132,-89148,297160,
%T -1002915,3421710,-11785890,40940460,-143291610,504932340,-1790214660,
%U 6382504440,-22870640910,82334307276,-297670187844,1080432533656,-3935861372604,14386251913656
%N a(n) = A184879(2*n, n) - A184879(2*n, n+1) where A184879(n, k) = Hypergeometric2F1(-2*k, 2*k-2*n, 1, -1) if 0<=k<=n.
%C Hankel transform is A184882.
%C Signed version of A007054. - _Philippe Deléham_, Mar 19 2014
%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of Its Properties</a>, J. Integer Sequences 4, No. 01.1.5, 2001
%H Fumitaka Yura, <a href="http://arxiv.org/abs/1411.6972">Hankel Determinant Solution for Elliptic Sequence</a>, arXiv:1411.6972 [nlin.SI], (25-November-2014); see p. 7
%F a(n) = 0^n + Sum_{k=0..2n} (C(2n,k)^2-C(2n+2,k)*C(2n-2,k))*(-1)^k.
%F G.f.: (8*x+1-sqrt(1+4*x)^3)/(2*x). - _Philippe Deléham_, Mar 19 2014
%F a(0) = 1, a(n) = (-1)^n*A007054(n-1) for n>0. - _Philippe Deléham_, Mar 19 2014
%F (n+1)*a(n) +2*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Nov 19 2014
%F a(n) = (-1)^n*A002421(n+1)/2 and 0 = a(n)*(+16*a(n+1) + 14*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n>0. - _Michael Somos_, Mar 13 2023
%e a(0) = 1;
%e a(1) = 1 - 4*1 = -3;
%e a(2) = 4*1 - 2 = 2;
%e a(3) = 5 - 4*2 = -3;
%e a(4) = 4*5 - 14 = 6;
%e a(5) = 42 - 4*14 = -14;
%e a(6) = 4*42 - 132 = 36;
%e a(7) = 429 - 4*132 = -99;
%e a(8) = 4*429 - 1430 = 286, etc; with A000108 = 1,1,2,5,14,42,132,429,1430, ... - _Philippe Deléham_, Mar 19 2014
%e G.f. = 1 - 3*x + 2*x^2 - 3*x^3 + 6*x^4 - 14*x^5 + 36*x^6 - 99*x^7 + ... - _Michael Somos_, Mar 13 2023
%p A184879 := proc(n,k) if k<0 or k >n then 0; else hypergeom([-2*k,2*k-2*n],[1],-1) ; simplify(%) ; end if; end proc:
%p A184881 := proc(n) A184879(2*n,n)-A184879(2*n,n+1) ; end proc:
%p seq(A184881(n),n=0..40) ; # _R. J. Mathar_, Feb 05 2011
%t h[n_, k_] := HypergeometricPFQ[{-2k, 2k - 2n}, {1}, -1];
%t a[0] = 1; a[n_] := h[2n, n] - h[2n, n + 1];
%t Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Nov 24 2017 *)
%Y Cf. A000108, A002421, A007054, A184879, A184882.
%K sign
%O 0,2
%A _Paul Barry_, Jan 24 2011
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