A035038 - OEIS

%I #42 Mar 20 2023 09:11:26

%S 0,0,0,0,0,0,1,8,37,130,386,1024,2510,5812,12911,27824,58651,121670,

%T 249528,507624,1026876,2069256,4158861,8344056,16721761,33486026,

%U 67025182,134116144,268313018,536724316,1073567387,2147277280,4294724471,8589650318,17179537972

%N a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,5).

%C Starting with "1", equals the eigensequence of a triangle with A000579 = binomial(n,6) = (1, 7, 28, 84, 210, ...) as the left column and the rest 1's. - _Gary W. Adamson_, Jul 24 2010

%H Alois P. Heinz, <a href="/A035038/b035038.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Eckhoff, <a href="http://dx.doi.org/10.1007/BF01297698">Der Satz von Radon in konvexen Produktstrukturen II</a>, Monat. f. Math., 73 (1969), 7-30.

%F From _Paul Barry_, Aug 23 2004: (Start)

%F G.f.: x^6/((1-2*x)*(1-x)^6).

%F a(n) = Sum_{k=0..n} C(n, k+6) = Sum_{k=6..n} C(n, k).

%F a(n) = 2*a(n-1) + C(n-1, 5).  (End)

%p a:= n-> (Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [8,-27,50,-55, 36,-13,2][i] else 0 fi)^(n))[1,7]:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 05 2008

%t Table[Sum[Binomial[n, k+6], {k,0,n}], {n,0,30}] (* _Zerinvary Lajos_, Jul 08 2009 *)

%t Table[2^n-Total[Binomial[n,Range[0,5]]],{n,0,40}] (* _Harvey P. Dale_, Oct 24 2017 *)

%o (Haskell)

%o a035038 n = a035038_list !! n

%o a035038_list = map (sum . drop 6) a007318_tabl

%o -- _Reinhard Zumkeller_, Jun 20 2015

%o (Magma) [n le 5 select 0 else (&+[Binomial(n,j): j in [6..n]]): n in [0..50]]; // _G. C. Greubel_, Mar 20 2023

%o (SageMath) [sum(binomial(n,j) for j in range(6,n+1)) for n in range(51)] # _G. C. Greubel_, Mar 20 2023

%Y Cf. A000079, A000225, A000295, A000579, A002663, A002664, A007318.

%Y Cf. A035039, A035040, A035041, A035042.

%K nonn,easy

%O 0,8

%A _N. J. A. Sloane_