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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.

%I A231622 #11 Aug 08 2018 22:21:55
%S A231622 2,-1,1,4,31,293,3326,44189,673471,11588884,222304897,4704612119,
%T A231622 108897613826,2737023412199,74236203425281,2161288643251828,
%U A231622 67228358271588991,2225173863019549229,78087247031912850686,2896042595237791161749,113184512236563589997407
%N 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.
%H A231622 G. C. Greubel, <a href="/A231622/b231622.txt">Table of n, a(n) for n = 0..403</a>
%F A231622 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.
%F A231622 a(-n) = a(n). a(n) = A003436(n) if n>1.
%F A231622 a(n) = (-1)^n*2*hypergeom([n, -n], [], 1/2). - _Peter Luschny_, Nov 10 2016
%e A231622 G.f. = 2 - x + x^2 + 4*x^3 + 31*x^4 + 293*x^5 + 3326*x^6 + 44189*x^7 + ...
%p A231622 A231622 := n -> (-1)^n*2*hypergeom([n, -n], [], 1/2):
%p A231622 seq(simplify(A231622(n)),n=0..19); # _Peter Luschny_, Nov 10 2016
%t A231622 a[ n_] := With[{m = Abs@n}, Boole[m == 0] + (2*m - 1)!! Hypergeometric1F1[ -m, 1 - 2*m, -2]]
%o A231622 (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))}
%Y A231622 Cf. A003436.
%K A231622 sign
%O A231622 0,1
%A A231622 _Michael Somos_, Nov 11 2013