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A002664 a(n) = 2^n - C(n,0)- ... - C(n,4).

%I A002664 M4395 N1851 #73 Mar 09 2025 12:28:39
%S A002664 0,0,0,0,0,1,7,29,93,256,638,1486,3302,7099,14913,30827,63019,127858,
%T A002664 258096,519252,1042380,2089605,4185195,8377705,16764265,33539156,
%U A002664 67090962,134196874,268411298,536843071,1073709893
%N A002664 a(n) = 2^n - C(n,0)- ... - C(n,4).
%C A002664 From _Gary W. Adamson_, Jul 24 2010: (Start)
%C A002664 Starting with "1" = eigensequence of a triangle with binomial C(n,5):
%C A002664 (1, 6, 21, 56, ...) as the left border and the rest 1's. (End)
%C A002664 The Kn26 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence minus the five leading zeros. - _Johannes W. Meijer_, Aug 15 2011
%C A002664 Starting (0, 0, 0, 0, 1, 7, 29, ...), this is the binomial transform of (0, 0, 0, 0, 1, 2, 2, 2, ...). Starting (1, 7, 29, ...), this is the binomial transform of (1, 6, 16, 26, 31, 32, 32, 32, ...). - _Gary W. Adamson_, Jul 28 2015
%D A002664 J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, Chapter 3, pp. 76-79.
%D A002664 J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.
%D A002664 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002664 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002664 Vincenzo Librandi, <a href="/A002664/b002664.txt">Table of n, a(n) for n = 0..1000</a>
%H A002664 R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H A002664 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 A002664 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002664 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A002664 H. P. Robinson, <a href="/A002664/a002664.pdf">Letter to N. J. A. Sloane, Mar 21 1985</a>
%H A002664 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,-20,30,-25,11,-2).
%F A002664 G.f.: x^5/((1-2*x)*(1-x)^5).
%F A002664 a(n) = Sum_{k=0..n} C(n, k+5) = Sum_{k=5..n} C(n, k); a(n) = 2a(n-1) + C(n-1, 4). - _Paul Barry_, Aug 23 2004
%F A002664 a(n) = 2^n - n^4/24 + n^3/12 - 11*n^2/24 - 7*n/12 - 1.  - _Bruno Berselli_, May 19 2011 [Robinson (1985) gives an alternative version of this formula, for a different offset. - _N. J. A. Sloane_, Oct 20 2015]
%F A002664 E.g.f.: exp(x)*(24*(exp(x) - 1) - 24*x - 12*x^2 - 4*x^3 - x^4)/24. - _Stefano Spezia_, Mar 09 2025
%p A002664 a:=n->sum(binomial(n+1,2*j),j=3..n+1): seq(a(n), n=0..30); # _Zerinvary Lajos_, May 12 2007
%p A002664 A002664:=1/(2*z-1)/(z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A002664 a=1;lst={};s1=s2=s3=s4=s5=0;Do[s1+=a;s2+=s1;s3+=s2;s4+=s3;s5+=s4;AppendTo[lst,s5];a=a*2,{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 10 2009 *)
%t A002664 Table[Sum[ Binomial[n, k + 5], {k, 0, n}], {n, 0, 30}] (* _Zerinvary Lajos_, Jul 08 2009 *)
%t A002664 Table[2^n-Total[Binomial[n,Range[0,4]]],{n,0,30}] (* or *) LinearRecurrence[ {7,-20,30,-25,11,-2},{0,0,0,0,0,1},40] (* _Harvey P. Dale_, Sep 03 2016 *)
%o A002664 (Magma) [2^n-n^4/24+n^3/12-11*n^2/24-7*n/12-1: n in [0..35]]; // _Vincenzo Librandi_, May 20 2011
%o A002664 (Haskell)
%o A002664 a002664 n = a002664_list !! n
%o A002664 a002664_list = map (sum . drop 5) a007318_tabl
%o A002664 -- _Reinhard Zumkeller_, Jun 20 2015
%Y A002664 a(n) = A055248(n, 5). Partial sums of A002663.
%Y A002664 Cf. A000079, A000225, A000295, A002662, A002663, A035038-A035042.
%Y A002664 Cf. A007318.
%K A002664 nonn,easy
%O A002664 0,7
%A A002664 _N. J. A. Sloane_