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

0, 0, 0, 0, 0, 1, 7, 29, 93, 256, 638, 1486, 3302, 7099, 14913, 30827, 63019, 127858, 258096, 519252, 1042380, 2089605, 4185195, 8377705, 16764265, 33539156, 67090962, 134196874, 268411298, 536843071, 1073709893

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Gary W. Adamson, Jul 24 2010: (Start)

Starting with "1" = eigensequence of a triangle with binomial C(n,5):

(1, 6, 21, 56, ...) as the left border and the rest 1's. (End)

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

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

References

• J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, Chapter 3, pp. 76-79.
• J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• R. K. Guy, Letter to N. J. A. Sloane
• Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• H. P. Robinson, Letter to N. J. A. Sloane, Mar 21 1985
• Index entries for linear recurrences with constant coefficients, signature (7,-20,30,-25,11,-2).

Crossrefs

a(n) = A055248(n, 5). Partial sums of A002663.

Cf. A000079, A000225, A000295, A002662, A002663, A035038-A035042.

Cf. A007318.

Programs

Haskell

a002664 n = a002664_list !! n
a002664_list = map (sum . drop 5) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

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

Maple

a:=n->sum(binomial(n+1,2*j),j=3..n+1): seq(a(n), n=0..30); # Zerinvary Lajos, May 12 2007
A002664:=1/(2*z-1)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)
Table[Sum[ Binomial[n, k + 5], {k, 0, n}], {n, 0, 30}] (* Zerinvary Lajos, Jul 08 2009 *)
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 *)

Formula

G.f.: x^5/((1-2*x)*(1-x)^5).

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

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]

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