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.
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
Keywords
Examples
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
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
Programs
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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
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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 *)
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
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