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%I M4152 N1725 #69 Sep 08 2022 08:44:31
%S 0,0,0,0,1,6,22,64,163,382,848,1816,3797,7814,15914,32192,64839,
%T 130238,261156,523128,1047225,2095590,4192510,8386560,16774891,
%U 33551806,67105912,134214424,268431773,536866822,1073737298
%N a(n) = 2^n - C(n,0) - C(n,1) - C(n,2) - C(n,3).
%C 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
%C 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
%C (1 + 6x + 22x^2 + 64x^3 + ...) = (1 + 3x + 6x^2 + 10x^3 + ...) * (1 + 3x + 7x^2 + 15x^3 + ...). - _Gary W. Adamson_, Mar 14 2012
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002663/b002663.txt">Table of n, a(n) for n = 0..1000</a>
%H J. Eckhoff, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002476398&IDDOC=247621">Der Satz von Radon in konvexen Productstrukturen II</a>, Monat. f. Math., 73 (1969), 7-30.
%H R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).
%F a(n) = 2^n - A000125(n).
%F G.f.: x^4/((1-2*x)*(1-x)^4). - _Simon Plouffe_ in his 1992 dissertation
%F a(n) = Sum_{k=0..n} binomial(n,k+4) = Sum_{k=4..n} binomial(n,k). - _Paul Barry_, Aug 23 2004
%F a(n) = 2*a(n-1) + binomial(n-1,3). - _Paul Barry_, Aug 23 2004
%F a(n) = (6*2^n - n^3 - 5*n - 6)/6. - _Mats Granvik_, _Gary W. Adamson_, Feb 17 2010
%p 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
%t 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 *)
%t Table[Sum[ Binomial[n + 4, k + 4], {k, 0, n}], {n, -4, 26}] (* _Zerinvary Lajos_, Jul 08 2009 *)
%o (Magma) [2^n - Binomial(n,0)- Binomial(n,1) - Binomial(n,2) - Binomial(n,3): n in [0..35]]; // _Vincenzo Librandi_, May 20 2011
%o (Haskell)
%o a002663 n = a002663_list !! n
%o a002663_list = map (sum . drop 4) a007318_tabl
%o -- _Reinhard Zumkeller_, Jun 20 2015
%o (PARI) a(n)=(6*2^n-n^3-5*n-6)/6 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y a(n)= A055248(n, 4). Partial sums of A002662.
%Y Cf. A000079, A000225, A000295, A002664, A035038-A035042, A007318.
%K nonn,easy
%O 0,6
%A _N. J. A. Sloane_
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