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

0, 3, 10, 22, 40, 65, 98, 140, 192, 255, 330, 418, 520, 637, 770, 920, 1088, 1275, 1482, 1710, 1960, 2233, 2530, 2852, 3200, 3575, 3978, 4410, 4872, 5365, 5890, 6448, 7040, 7667, 8330, 9030, 9768, 10545, 11362, 12220, 13120, 14063, 15050, 16082, 17160, 18285

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Rowan Rowlands, Sheila Sundaram, and Lei Xue, Topology of Cut Complexes of Graphs, arXiv:2304.13675 [math.CO], 2023.
• G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
• G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48.
• Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
• P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
• P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
• 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 (4,-6,4,-1).

Crossrefs

a(n) = A095660(n+2, 3): fourth column of (1, 3)-Pascal triangle.

Cf. A000027, A000096, A006504.

Cf. A000292, A002378.

Row n=3 of A144064.

Programs

Maple

A006503:=-(-3+2*z)/(z-1)**4; # [Simon Plouffe in his 1992 dissertation.]

Mathematica

Clear["Global`*"] a[n_] := n(n + 1)(n + 8)/3! Do[Print[n, " ", a[n]], {n, 1, 25}] (* Sergio Falcon, May 22 2008 *)
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 *)

PARI

x='x+O('x^50); concat([0], Vec(x*(3-2*x)/(1-x)^4)) \\ G. C. Greubel, May 11 2017

Formula

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

G.f.: x*(3-2*x)/(1-x)^4.

a(n) = A000292(n) + A002378(n). - Reinhard Zumkeller, Sep 24 2008

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

Extensions

Better description from Jeffrey Shallit, Aug 1995