A231622 (2*n+1)*a(n+1) = (4*n^2+1)*a(n) + (2*n+1)*a(n-1) with n>1, a(0)=2, a(1)=-1.

2, -1, 1, 4, 31, 293, 3326, 44189, 673471, 11588884, 222304897, 4704612119, 108897613826, 2737023412199, 74236203425281, 2161288643251828, 67228358271588991, 2225173863019549229, 78087247031912850686, 2896042595237791161749, 113184512236563589997407

Offset: 0

Author

Michael Somos, Nov 11 2013

Keywords

• sign

Examples

			G.f. = 2 - x + x^2 + 4*x^3 + 31*x^4 + 293*x^5 + 3326*x^6 + 44189*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..403

Crossrefs

Cf. A003436.

Programs

Maple

A231622 := n -> (-1)^n*2*hypergeom([n, -n], [], 1/2):
seq(simplify(A231622(n)),n=0..19); # Peter Luschny, Nov 10 2016

Mathematica

a[ n_] := With[{m = Abs@n}, Boole[m == 0] + (2*m - 1)!! Hypergeometric1F1[ -m, 1 - 2*m, -2]]

PARI

{a(n) = n=abs(n); if( n<2, 2 - 3*(n>0), ( a(n-1) * (4*n^2 - 8*n + 5) + a(n-2) * (2*n-1) ) / (2*n-3))}

Formula

E.g.f. A(x) satisfies 0 = f(A(x), A'(x), A''(x)) where f(u0, u1, u2) = (3 + 2*x)*u0 + (5 + 2*x)*u1 + (-1 + 4*x^2)*u2.

a(-n) = a(n). a(n) = A003436(n) if n>1.

a(n) = (-1)^n*2*hypergeom([n, -n], [], 1/2). - Peter Luschny, Nov 10 2016