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A002664
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2^n - C(n,0)- ... - C(n,4).
(Formerly M4395 N1851)
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13
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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
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OFFSET
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0,7
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COMMENTS
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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]
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REFERENCES
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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).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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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
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MAPLE
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a:=n->sum(binomial(n+1, 2*j), j=3..n+1): seq(a(n), n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2007
A002664:=1/(2*z-1)/(z-1)**5; [Conjectured by Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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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 (zerinvarylajos(AT)yahoo.com), Jul 08 2009 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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