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A002664 2^n - C(n,0)- ... - C(n,4).
(Formerly M4395 N1851)
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Contribution 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]

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, Chapter 3, p.s 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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

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 (* From Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)

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

PROG

(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

CROSSREFS

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

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

Sequence in context: A001779 A053295 A055798 * A042609 A002941 A193655

Adjacent sequences:  A002661 A002662 A002663 * A002665 A002666 A002667

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 25 08:09 EDT 2014. Contains 248518 sequences.