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A006503 a(n) = n*(n+1)*(n+8)/6.

%I A006503 M2835 #70 May 07 2025 09:03:05
%S A006503 0,3,10,22,40,65,98,140,192,255,330,418,520,637,770,920,1088,1275,
%T A006503 1482,1710,1960,2233,2530,2852,3200,3575,3978,4410,4872,5365,5890,
%U A006503 6448,7040,7667,8330,9030,9768,10545,11362,12220,13120,14063,15050,16082,17160,18285
%N A006503 a(n) = n*(n+1)*(n+8)/6.
%C A006503 If Y is a 3-subset of an n-set X then, for n>=4, a(n-4) is the number of 3-subsets of X having at most one element in common with Y. - _Milan Janjic_, Nov 23 2007
%C A006503 The coefficient of x^3 in (1-x-x^2)^{-n} is the coefficient of x^3 in (1+x+2x^2+3x^3)^n. Using the multinomial theorem one then finds that a(n)=n(n+1)(n+8)/3!. - _Sergio Falcon_, May 22 2008
%D A006503 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D A006503 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006503 G. C. Greubel, <a href="/A006503/b006503.txt">Table of n, a(n) for n = 0..5000</a>
%H A006503 Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Rowan Rowlands, Sheila Sundaram, and Lei Xue, <a href="https://arxiv.org/abs/2304.13675">Topology of Cut Complexes of Graphs</a>, arXiv:2304.13675 [math.CO], 2023.
%H A006503 G. E. Bergum and V. E. Hoggatt, Jr., <a href="/A006503/a006503.pdf">Numerator polynomial coefficient array for the convolved Fibonacci sequence</a>, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
%H A006503 G. E. Bergum and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-1/bergum.pdf">Numerator polynomial coefficient array for the convolved Fibonacci sequence</a>, Fib. Quart., 14 (1976), 43-48.
%H A006503 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
%H A006503 P. Moree, <a href="https://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H A006503 P. Moree, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Moree/moree12.htm">Convoluted Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
%H A006503 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 A006503 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006503 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A006503 a(n) = n*(n+1)*(n+8)/6.
%F A006503 G.f.: x*(3-2*x)/(1-x)^4.
%F A006503 a(n) = A000292(n) + A002378(n). - _Reinhard Zumkeller_, Sep 24 2008
%F A006503 a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4) with a(0)=0, a(1)=3, a(2)=10, a(3)=22. - _Harvey P. Dale_, Jan 27 2016
%p A006503 A006503:=-(-3+2*z)/(z-1)**4; # [_Simon Plouffe_ in his 1992 dissertation.]
%t A006503 Clear["Global`*"] a[n_] := n(n + 1)(n + 8)/3! Do[Print[n, " ", a[n]], {n, 1, 25}] (* _Sergio Falcon_, May 22 2008 *)
%t A006503 Table[n(n+1)(n+8)/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,3,10,22},50] (* _Harvey P. Dale_, Jan 27 2016 *)
%o A006503 (PARI) x='x+O('x^50); concat([0], Vec(x*(3-2*x)/(1-x)^4)) \\ _G. C. Greubel_, May 11 2017
%Y A006503 a(n) = A095660(n+2, 3): fourth column of (1, 3)-Pascal triangle.
%Y A006503 Cf. A000027, A000096, A006504.
%Y A006503 Cf. A000292, A002378.
%Y A006503 Row n=3 of A144064.
%K A006503 nonn,easy
%O A006503 0,2
%A A006503 _N. J. A. Sloane_
%E A006503 Better description from _Jeffrey Shallit_, Aug 1995