A002663 a(n) = 2^n - C(n,0) - C(n,1) - C(n,2) - C(n,3). (Formerly M4152 N1725)

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

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

(1 + 6x + 22x^2 + 64x^3 + ...) = (1 + 3x + 6x^2 + 10x^3 + ...) * (1 + 3x + 7x^2 + 15x^3 + ...). - Gary W. Adamson, Mar 14 2012

The sequence starting (1, 6, 22, ...) is the binomial transform of A171418 and starting (0, 0, 0, 1, 6, 22, ...) is the binomial transform of (0, 0, 0, 1, 2, 2, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015

References

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

J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.

R. K. Guy, Letter to N. J. A. Sloane

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

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Formula

a(n) = 2^n - A000125(n).

G.f.: x^4/((1-2*x)*(1-x)^4). - Simon Plouffe in his 1992 dissertation

a(n) = Sum_{k=0..n} binomial(n,k+4) = Sum_{k=4..n} binomial(n,k). - Paul Barry, Aug 23 2004

a(n) = 2*a(n-1) + binomial(n-1,3). - Paul Barry, Aug 23 2004

a(n) = (6*2^n - n^3 - 5*n - 6)/6. - Mats Granvik, Gary W. Adamson, Feb 17 2010

Maple

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 (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Table[Sum[ Binomial[n + 4, k + 4], {k, 0, n}], {n, -4, 26}] (* Zerinvary Lajos, 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
(Haskell)
a002663 n = a002663_list !! n
a002663_list = map (sum . drop 4) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015
(PARI) a(n)=(6*2^n-n^3-5*n-6)/6 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

a(n)= A055248(n, 4). Partial sums of A002662.

Cf. A000079, A000225, A000295, A002664, A035038-A035042, A007318.

Sequence in context: A280481 A055797 A001925 * A099855 A347435 A003469

Adjacent sequences: A002660 A002661 A002662 * A002664 A002665 A002666

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved