|
| |
|
|
A002663
|
|
2^n - C(n,0)- ... - C(n,3).
(Formerly M4152 N1725)
|
|
16
| |
|
|
0, 0, 0, 0, 1, 6, 22, 64, 163, 382, 848, 1816, 3797, 7814, 15914, 32192, 64839, 130238, 261156, 523128, 1047225, 2095590, 4192510, 8386560, 16774891, 33551806, 67105912, 134214424, 268431773, 536866822, 1073737298
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,6
|
|
|
COMMENTS
| Starting with "1" = eigensequence of a triangle with bin(n,4), A000332 as the left border: (1, 5, 15, 35, 70,...) and the rest 1's. [Gary W. Adamson, Jul 24 2010]
The Kn25 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence minus the four leading zeros. [Johannes W. Meijer, Aug 14 2011]
|
|
|
REFERENCES
| 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
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
|
|
|
FORMULA
| G.f.: x^4/((1-2*x)*(1-x)^4).
a(n)=sum{k=0..n, C(n,k+4)} = sum{k=4..n, C(n,k)}; a(n) = 2*a(n-1) + C(n-1,3). - Paul Barry, Aug 23 2004
a(n)=1/6*(-n^3-9*n^2-32*n+3*2^(n+4)-48) [From Mats Granvik, Gary W. Adamson, Feb 17 2010]
|
|
|
MAPLE
| A002663 := -1/(2*z-1)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
A002663 := proc(n): 2^n - add(binomial(n, k), k=0..3) end: seq(A002663(n), n=0..30); # [Johannes W. Meijer, Aug 14 2011]
|
|
|
MATHEMATICA
| a=1; lst={}; s1=s2=s3=s4=0; Do[s1+=a; s2+=s1; s3+=s2; s4+=s3; AppendTo[lst, s4]; a=a*2, {n, 5!}]; lst [From Vladimir Orlovsky, Jan 10 2009]
Table[Sum[ Binomial[n + 4, k + 4], {k, 0, n}], {n, -4, 26}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
|
|
|
PROG
| (MAGMA) [2^n - Binomial(n, 0)- Binomial(n, 1) - Binomial(n, 2) - Binomial(n, 3): n in [0..35]]; // Vincenzo Librandi, May 20 2011
|
|
|
CROSSREFS
| a(n)= A055248(n, 4). Partial sums of A002662.
Cf. A000079, A000225, A000295, A002662, A002664, A035038-A035042.
Sequence in context: A053739 A055797 A001925 * A099855 A003469 A189418
Adjacent sequences: A002660 A002661 A002662 * A002664 A002665 A002666
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
