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URL: https://oeis.org/A184881

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A184881
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.
3
1, -3, 2, -3, 6, -14, 36, -99, 286, -858, 2652, -8398, 27132, -89148, 297160, -1002915, 3421710, -11785890, 40940460, -143291610, 504932340, -1790214660, 6382504440, -22870640910, 82334307276, -297670187844, 1080432533656, -3935861372604, 14386251913656
OFFSET
0,2
COMMENTS
Hankel transform is A184882.
Signed version of A007054. - Philippe Deléham, Mar 19 2014
LINKS
J. W. Layman, The Hankel Transform and Some of Its Properties, J. Integer Sequences 4, No. 01.1.5, 2001
Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], (25-November-2014); see p. 7
FORMULA
a(n) = 0^n + Sum_{k=0..2n} (C(2n,k)^2-C(2n+2,k)*C(2n-2,k))*(-1)^k.
G.f.: (8*x+1-sqrt(1+4*x)^3)/(2*x). - Philippe Deléham, Mar 19 2014
a(0) = 1, a(n) = (-1)^n*A007054(n-1) for n>0. - Philippe Deléham, Mar 19 2014
(n+1)*a(n) +2*(2*n-3)*a(n-1)=0. - R. J. Mathar, Nov 19 2014
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
EXAMPLE
a(0) = 1;
a(1) = 1 - 4*1 = -3;
a(2) = 4*1 - 2 = 2;
a(3) = 5 - 4*2 = -3;
a(4) = 4*5 - 14 = 6;
a(5) = 42 - 4*14 = -14;
a(6) = 4*42 - 132 = 36;
a(7) = 429 - 4*132 = -99;
a(8) = 4*429 - 1430 = 286, etc; with A000108 = 1,1,2,5,14,42,132,429,1430, ... - Philippe Deléham, Mar 19 2014
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
MAPLE
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:
A184881 := proc(n) A184879(2*n, n)-A184879(2*n, n+1) ; end proc:
seq(A184881(n), n=0..40) ; # R. J. Mathar, Feb 05 2011
MATHEMATICA
h[n_, k_] := HypergeometricPFQ[{-2k, 2k - 2n}, {1}, -1];
a[0] = 1; a[n_] := h[2n, n] - h[2n, n + 1];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 24 2017 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Paul Barry, Jan 24 2011
STATUS
approved