%I #37 Apr 08 2020 12:52:09

%S 0,0,0,0,0,0,0,1,9,46,176,562,1586,4096,9908,22819,50643,109294,

%T 230964,480492,988116,2014992,4084248,8243109,16587165,33308926,

%U 66794952,133820134,267936278,536249296,1072973612,2146540999

%N a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,6).

%C Partial sums of A035038.

%H Reinhard Zumkeller, <a href="/A035039/b035039.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.

%H Ângela Mestre, José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (9,-35,77,-105,91,-49,15,-2).

%F a(n) = A055248(n,7).

%F G.f.: x^7/((1-2*x)*(1-x)^7).

%F a(n) = Sum_{k=0..n}, C(n, k+7) = Sum_{k=7..n} C(n, k); a(n) = 2a(n-1) + C(n-1, 6). - _Paul Barry_, Aug 23 2004

%p a:=n->sum(binomial(n,j),j=7..n): seq(a(n), n=0..31); # _Zerinvary Lajos_, Feb 12 2007

%t a=1;lst={};s1=s2=s3=s4=s5=s6=s7=0;Do[s1+=a;s2+=s1;s3+=s2;s4+=s3;s5+=s4;s6+=s5;s7+=s6;AppendTo[lst,s7];a=a*2,{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 10 2009 *)

%t Table[2^n-Total[Binomial[n,Range[0,6]]],{n,40}] (* or *) LinearRecurrence[ {9,-35,77,-105,91,-49,15,-2},{0,0,0,0,0,0,0,1},40] (* _Harvey P. Dale_, Apr 22 2016 *)

%o (Haskell)

%o a035039 n = a035039_list !! n

%o a035039_list = map (sum . drop 7) a007318_tabl

%o -- _Reinhard Zumkeller_, Jun 20 2015

%Y Cf. A000079, A000225, A000295, A002663, A002664, A035038-A035042.

%Y Cf. A007318.

%K nonn,easy

%O 0,9

%A _N. J. A. Sloane_