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A025178 First differences of the central trinomial coefficients A002426. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
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)
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 *)
PROG
(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
CROSSREFS
Sequence in context: A028860 A152035 A026151 * A231295 A087211 A161177
KEYWORD
nonn,easy
AUTHOR
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
STATUS
approved

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Last modified June 14 02:53 EDT 2024. Contains 373392 sequences. (Running on oeis4.)