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A054109 a(n) = T(2*n+1, n), array T as in A054106.

%I A054109 #39 Jan 18 2020 10:57:16
%S A054109 1,2,8,27,99,363,1353,5082,19228,73150,279566,1072512,4127788,
%T A054109 15930512,61628248,238911947,927891163,3609676487,14062955413,
%U A054109 54860308997,214268628223,837780853637,3278934510163,12844867331387
%N A054109 a(n) = T(2*n+1, n), array T as in A054106.
%C A054109 Hankel transform of A054109. - _Paul Barry_, Nov 04 2009
%C A054109 From _Paul Barry_, Mar 29 2010: (Start)
%C A054109 Hankel transform is A167478 (correction of previous entry).
%C A054109 The aerated sequence 0,0,1,0,2,0,8,0,... has e.g.f. Integral_{t=0..x} cos(x-t)*Bessel_I(1,2t). (End)
%C A054109 Hankel transform of 0,1,2,8,27,... is -F(2n). - _Paul Barry_, Jan 17 2020
%H A054109 Vincenzo Librandi, <a href="/A054109/b054109.txt">Table of n, a(n) for n = 0..1000</a>
%F A054109 a(n-1) = (1/2)*(-1)^n*Sum_{k=1..n} (-1)^k*binomial(2k, k). - _Benoit Cloitre_, Nov 07 2002
%F A054109 Conjecture: (n+1)*a(n) + (-3*n-1)*a(n-1) + 2*(-2*n-1)*a(n-2) = 0. - _R. J. Mathar_, Nov 24 2012
%F A054109 a(n) ~ 2^(2*n+3) / (5*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 12 2014
%F A054109 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2k+1,k+1). - _Paul Barry_, Jan 17 2020
%F A054109 G.f.: c(x)B(x)/(1+x), c(x) g.f. of A000108, B(x) g.f. of A000984. - _Paul Barry_, Jan 17 2020
%F A054109 a(n) = binomial(2*n+3, n+2)*hypergeom([1, n+5/2], [n+3], -4) + (-1)^n*(5 - sqrt(5)) /10. - _Peter Luschny_, Jan 18 2020
%p A054109 a := n -> abs(add(binomial(-j-1, -2*j-2), j=0..n)):
%p A054109 seq(a(n), n=0..23); # _Zerinvary Lajos_, Oct 03 2007
%p A054109 gf := ((1 - 4*x)^(-1/2) - 1)/(2*x*(x + 1)): ser := series(gf, x, 32):
%p A054109 seq(coeff(ser, x, n), n=0..23); # _Peter Luschny_, Jan 18 2020
%t A054109 Table[FullSimplify[1/2*(-1)^(1+n) * (-1+1/Sqrt[5]-(-1)^n*Binomial[2*(2+n), 2+n] * Hypergeometric2F1[1, 5/2+n, 3+n, -4])],{n,0,20}] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%t A054109 Table[1/2*(-1)^(n+1)*Sum[(-1)^k*Binomial[2*k, k],{k,1,n+1}],{n,0,20}] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%o A054109 (PARI) a(n)=(1/2)*(-1)^(n+1)*sum(k=1,n+1,(-1)^k*binomial(2*k,k))
%Y A054109 Cf. A054106, A000108, A000984.
%K A054109 nonn
%O A054109 0,2
%A A054109 _Clark Kimberling_