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

Author

Paul Barry, Jan 24 2011

Keywords

• sign

Comments

Hankel transform is A184882.

Signed version of A007054. - Philippe Deléham, Mar 19 2014

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

Crossrefs

Cf. A000108, A002421, A007054, A184879, A184882.

Programs

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 *)

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