|
%I #28 Feb 26 2024 10:10:56
%S 1,6,36,224,1440,9504,64064,439296,3055104,21498880,152807424,
%T 1095450624,7911587840,57511157760,420459724800,3089600348160,
%U 22806128885760,169033661153280,1257467341701120,9385880636620800,70271680244613120,527595313582571520
%N Expansion of g.f. 8/(1+sqrt(1-8*x))^3.
%C 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]
%F a(n) = 6(n+1)*2^(n-2)*Cat(n+2)/(2n+3), where Cat(n)=A000108(n). - _Ralf Stephan_, Mar 11 2004
%F 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
%F a(n) = (n+1) * A000257(n+1). - _F. Chapoton_, Feb 26 2024
%F 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
%F From _Amiram Eldar_, Mar 24 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 22/49 + 808*arcsin(1/(2*sqrt(2)))/(147*sqrt(7)).
%F Sum_{n>=0} (-1)^n/a(n) = 26/81 + 376*arcsinh(1/(2*sqrt(2)))/243. (End)
%t CoefficientList[Series[8/(1 + Sqrt[1 - 8*x])^3, {x, 0, 21}], x] (* _Amiram Eldar_, Mar 24 2022 *)
%Y Cf. A002421, A000108, A000257.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jul 13 2003
| 