A115902 Expansion of (1-8*x)^(-3/2).

1, 12, 120, 1120, 10080, 88704, 768768, 6589440, 56010240, 472975360, 3972993024, 33228668928, 276905574400, 2300446310400, 19060840857600, 157569617756160, 1299949346488320, 10705465206374400, 88022713919078400, 722712809019801600, 5926245033962373120

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Youngja Park and SeungKyung Park, Enumeration of generalized lattice paths by string types, peaks, and ascents, Discrete Mathematics 339.11 (2016): 2652-2659.

Formula

G.f.: 1/((1-8*x)*sqrt(1-8*x)) = 1F0(3/2;;8x).

a(n) = Jacobi_P(n,1/2,1/2,1)*8^n.

a(n) = 2^n*(2*n+1)*binomial(2*n,n).

a(n) = (2*n+1)*A059304(n).

a(n) = 2^n*A002457(n).

D-finite with recurrence: n*a(n) -4*(2*n+1)*a(n-1) =0. - R. J. Mathar, Nov 14 2011

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 8*x*(2*k+3)/(8*x*(2*k+3) + 2*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

Sum_{n>=0} (-1)^n/a(n) = 4/3*log(2). - Daniel Suteu, Oct 31 2017

Sum_{n>=0} 1/a(n) = 8*arcsin(1/sqrt(8))/sqrt(7). - Amiram Eldar, Jan 27 2024

Maple

a:= n-> add((binomial(2*n, n))*2^(n-2), j=1..n): seq(a(n), n=1..20); # Zerinvary Lajos, May 03 2007

Mathematica

CoefficientList[Series[(1-8x)^-(3/2), {x, 0, 30}], x] (* Harvey P. Dale, Jul 13 2012 *)

Crossrefs

Cf. A002457, A059304.

Sequence in context: A012273 A008465 A291391 * A277491 A004332 A129329

Adjacent sequences: A115899 A115900 A115901 * A115903 A115904 A115905

Keyword

easy,nonn

Author

Paul Barry, Feb 02 2006

Status

approved