A025178 First differences of the central trinomial coefficients A002426.

0, 2, 4, 12, 32, 90, 252, 714, 2032, 5814, 16700, 48136, 139152, 403286, 1171380, 3409020, 9938304, 29017878, 84844044, 248382516, 727971360, 2135784798, 6272092596, 18435108258, 54228499920, 159636389850, 470256930052, 1386170197704

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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."

Note that n-1 divides a(n) for n>=2. - T. D. Noe, Mar 16 2005

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001006, A002426, A005717, A007971, A025177, A079309, A097893, A105696, A162551.

Programs

Maple

a := n -> 2*(n-1)*hypergeom([1-n/2, 3/2-n/2], [2], 4):
seq(simplify(a(n)), n=1..28); # Peter Luschny, Oct 29 2015

Mathematica

Rest[Differences[CoefficientList[Series[x/Sqrt[1-2x-3x^2],{x,0,30}],x]]] (* Harvey P. Dale, Aug 22 2011 *)
Differences[Table[Hypergeometric2F1[(1-n)/2,1-n/2,1,4],{n,1,29}]] (* Peter Luschny, Nov 03 2015 *)

PARI

a(n) = sum(k=1, n\2, binomial(n-1,2*k-1)*binomial(2*k,k)); \\ Altug Alkan, Oct 29 2015

Sage

def a():
    b, c, n = 0, 2, 2
    yield b
    while True:
        yield c
        b, c = c, ((2*n-1)*c+3*(n-1)*b)*n//((n+1)*(n-1))
        n += 1
A025178 = a()
print([next(A025178) for  in (1..20)]) # _Peter Luschny, Nov 04 2015

Formula

a(n) = T(n,n) for n>=1, where T is the array defined in A025177.

a(n) = A002426(n+1) - A002426(n). - Benoit Cloitre, Nov 02 2002

a(n) is asymptotic to c*3^n/sqrt(n) with c around 1.02... - Benoit Cloitre, Nov 02 2002

a(n) = 2*(n-1)*A001006(n-2). - M. F. Hasler, Oct 24 2011

a(n) = 2*A005717(n-1). - R. J. Mathar, Jul 09 2012

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

G.f.: -1/x + (1/x-1)/sqrt(1-2*x-3*x^2). - Sergei N. Gladkovskii, Aug 16 2012

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]

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

From Peter Bala, Oct 28 2015: (Start)

a(n) = Sum_{k = 0..floor(n/2)} binomial(n-1,2*k-1)*binomial(2*k,k). Cf. A097893.

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)

From Peter Luschny, Oct 29 2015: (Start)

a(n) = 2*(n-1)*hypergeom([1-n/2,3/2-n/2],[2],4).

a(n) = (n-1)!*[x^(n-1)](2*exp(x)*BesselI(1,2*x)).

a(n) = (n-1)*A007971(n) for n>=2.

A105696(n) = a(n-1) + a(n) for n>=2.

A162551(n-2) = (1/2)*Sum_{k=1..n} binomial(n,k)*a(k) for n>=2.

A079309(n) = (1/2)*Sum_{k=1..2*n} (-1)^k*binomial(2*n,k)*a(k) for n>=1.

(End)

Extensions

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