A006503 a(n) = n*(n+1)*(n+8)/6. (Formerly M2835)

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

0,2

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.

M. Janjic, 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).

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

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 *)

Prog

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

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.

Sequence in context: A346166 A122795 A140066 * A248851 A023554 A294414

Adjacent sequences: A006500 A006501 A006502 * A006504 A006505 A006506

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Better description from Jeffrey Shallit, Aug 1995

Status

approved