A127040 - OEIS

%I #22 Dec 29 2023 11:49:17

%S 1,1,5,6,28,36,165,220,1001,1365,6188,8568,38760,54264,245157,346104,

%T 1562275,2220075,10015005,14307150,64512240,92561040,417225900,

%U 600805296,2707475148,3910797436,17620076360,25518731280,114955808528

%N a(n) = binomial(floor((3n+4)/2),floor(n/2)).

%C With offset 2, the number of compositions of n into floor(n/2) parts, which is an upper bound for A007874.

%H T. D. Noe, <a href="/A127040/b127040.txt">Table of n, a(n) for n=0..200</a>

%F From _Benedict W. J. Irwin_, Aug 16 2016: (Start)

%F G.f.: (-1 + (2*cos(arcsin(3*sqrt(3)*x/2)/3))/sqrt(4-27*x^2) + 3*x^3*2F1(4/3,5/3;5/2;27*x^2/4))/(3*x^2).

%F E.g.f.: 2F3(4/3,5/3;1/2,3/2,2;27*x^2/16) + x*2F3(4/3,5/3;1,3/2,5/2;27*x^2/16).

%F (End)

%F D-finite with recurrence 8*(n+2)*(n+1)*a(n) -84*(n-1)*(n+1)*a(n-1) +6*(-33*n^2+54*n-8)*a(n-2) +9*(63*n^2-63*n-16)*a(n-3) +108*(3*n-5)*(3*n-7)*a(n-4)=0. - _R. J. Mathar_, Feb 08 2021

%p seq(sum(binomial(n+k, k-1), k=0..ceil((n+1)/2)), n=0..28); # _Zerinvary Lajos_, Apr 11 2007

%t CoefficientList[Series[(-1 + (2 Cos[1/3 ArcSin[(3 Sqrt[3] x)/2]])/Sqrt[4 - 27 x^2] + 3 x^3 Hypergeometric2F1[4/3, 5/3, 5/2, (27 x^2)/4])/(3 x^2), {x, 0, 20}], x] (* _Benedict W. J. Irwin_, Aug 16 2016 *)

%t Table[Binomial[Floor[(3 n + 4)/2], Floor[n/2]], {n, 0, 28}] (* _Michael De Vlieger_, Aug 18 2016 *)

%o (PARI) a(n) = binomial((3*n+4)\2, n\2); \\ _Michel Marcus_, Sep 09 2016

%Y Cf. A004319 (bisection), A025174 (bisection), A099578.

%K nonn

%O 0,3

%A _T. D. Noe_, Jan 03 2007