A085687 Expansion of g.f. 8/(1+sqrt(1-8*x))^3.

1, 6, 36, 224, 1440, 9504, 64064, 439296, 3055104, 21498880, 152807424, 1095450624, 7911587840, 57511157760, 420459724800, 3089600348160, 22806128885760, 169033661153280, 1257467341701120, 9385880636620800, 70271680244613120, 527595313582571520

Offset

0,2

Comments

a(n) is also the number of paths of length 2(n+1) in a binary tree between two vertices that are 2 steps apart. [David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010]

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = 6(n+1)*2^(n-2)*Cat(n+2)/(2n+3), where Cat(n)=A000108(n). - Ralf Stephan, Mar 11 2004

G.f.: c(2x)^3, where c(x) is the g.f. of A000108; a(n)=3(n+1)2^n*Cat(n+1)/(n+3); - Paul Barry, Dec 08 2004

a(n) = (n+1) * A000257(n+1). - F. Chapoton, Feb 26 2024

D-finite with recurrence: (n+3)*a(n) -2*(5*n+7)*a(n-1) +8*(2*n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2011

From Amiram Eldar, Mar 24 2022: (Start)

Sum_{n>=0} 1/a(n) = 22/49 + 808*arcsin(1/(2*sqrt(2)))/(147*sqrt(7)).

Sum_{n>=0} (-1)^n/a(n) = 26/81 + 376*arcsinh(1/(2*sqrt(2)))/243. (End)

Mathematica

CoefficientList[Series[8/(1 + Sqrt[1 - 8*x])^3, {x, 0, 21}], x] (* Amiram Eldar, Mar 24 2022 *)

Crossrefs

Cf. A002421, A000108, A000257.

Sequence in context: A351056 A166748 A200378 * A242136 A129327 A244889

Adjacent sequences: A085684 A085685 A085686 * A085688 A085689 A085690

Keyword

nonn

Author

N. J. A. Sloane, Jul 13 2003

Status

approved