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%I #76 Feb 26 2020 07:16:53
%S 0,2,4,12,32,90,252,714,2032,5814,16700,48136,139152,403286,1171380,
%T 3409020,9938304,29017878,84844044,248382516,727971360,2135784798,
%U 6272092596,18435108258,54228499920,159636389850,470256930052,1386170197704
%N First differences of the central trinomial coefficients A002426.
%C Previous name was: "a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0 = s(n), |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array defined in A025177."
%C Note that n-1 divides a(n) for n>=2. - _T. D. Noe_, Mar 16 2005
%H G. C. Greubel, <a href="/A025178/b025178.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = T(n,n) for n>=1, where T is the array defined in A025177.
%F a(n) = A002426(n+1) - A002426(n). - _Benoit Cloitre_, Nov 02 2002
%F a(n) is asymptotic to c*3^n/sqrt(n) with c around 1.02... - _Benoit Cloitre_, Nov 02 2002
%F a(n) = 2*(n-1)*A001006(n-2). - _M. F. Hasler_, Oct 24 2011
%F a(n) = 2*A005717(n-1). - _R. J. Mathar_, Jul 09 2012
%F E.g.f. Integral(Integral(2*exp(x)*((1-1/x)*BesselI(1,2*x) + 2*BesselI(0,2*x)))). - _Sergei N. Gladkovskii_, Aug 16 2012
%F G.f.: -1/x + (1/x-1)/sqrt(1-2*x-3*x^2). - _Sergei N. Gladkovskii_, Aug 16 2012
%F D-finite with recurrence: a(n) = ((2+n)*a(n-2)+3*(3-n)*a(n-3)+3*(n-1)*a(n-1))/n, a(0)=1, a(1)=0, a(2)=2. - _Sergei N. Gladkovskii_, Aug 16 2012 [adapted to new offset by _Peter Luschny_, Nov 04 2015]
%F G.f.: (1-x)/x^2*G(0) - 1/x^2 , where G(k)= 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 06 2013
%F From _Peter Bala_, Oct 28 2015: (Start)
%F a(n) = Sum_{k = 0..floor(n/2)} binomial(n-1,2*k-1)*binomial(2*k,k). Cf. A097893.
%F n*(n-2)*a(n) = (2*n-3)*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2) with a(1) = 0, a(2) = 2. (End)
%F From _Peter Luschny_, Oct 29 2015: (Start)
%F a(n) = 2*(n-1)*hypergeom([1-n/2,3/2-n/2],[2],4).
%F a(n) = (n-1)!*[x^(n-1)](2*exp(x)*BesselI(1,2*x)).
%F a(n) = (n-1)*A007971(n) for n>=2.
%F A105696(n) = a(n-1) + a(n) for n>=2.
%F A162551(n-2) = (1/2)*Sum_{k=1..n} binomial(n,k)*a(k) for n>=2.
%F A079309(n) = (1/2)*Sum_{k=1..2*n} (-1)^k*binomial(2*n,k)*a(k) for n>=1.
%F (End)
%p a := n -> 2*(n-1)*hypergeom([1-n/2, 3/2-n/2], [2], 4):
%p seq(simplify(a(n)), n=1..28); # _Peter Luschny_, Oct 29 2015
%t Rest[Differences[CoefficientList[Series[x/Sqrt[1-2x-3x^2],{x,0,30}],x]]] (* _Harvey P. Dale_, Aug 22 2011 *)
%t Differences[Table[Hypergeometric2F1[(1-n)/2,1-n/2,1,4],{n,1,29}]] (* _Peter Luschny_, Nov 03 2015 *)
%o (PARI) a(n) = sum(k=1, n\2, binomial(n-1,2*k-1)*binomial(2*k,k)); \\ _Altug Alkan_, Oct 29 2015
%o (Sage)
%o def a():
%o b, c, n = 0, 2, 2
%o yield b
%o while True:
%o yield c
%o b, c = c, ((2*n-1)*c+3*(n-1)*b)*n//((n+1)*(n-1))
%o n += 1
%o A025178 = a()
%o print([next(A025178) for _ in (1..20)]) # _Peter Luschny_, Nov 04 2015
%Y Cf. A001006, A002426, A005717, A007971, A025177, A079309, A097893, A105696, A162551.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_
%E New name based on a comment by _T. D. Noe_, Mar 16 2005, offset set to 1 and a(1) = 0 prepended by _Peter Luschny_, Nov 04 2015
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