A005288 a(n) = C(n,5) + C(n,4) - C(n,3) + 1, n >= 7. (Formerly M3090)

3, 22, 71, 169, 343, 628, 1068, 1717, 2640, 3914, 5629, 7889, 10813, 14536, 19210, 25005, 32110, 40734, 51107, 63481, 78131, 95356, 115480, 138853, 165852, 196882, 232377, 272801, 318649, 370448, 428758, 494173, 567322

Offset

6,1

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 15.

E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=6..38.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241-242. (Annotated scanned copy)

R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988

R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.

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,-15,20,-15,6,-1).

Formula

a(n) = C(n+3, 5) - C(n+2, 3) + C(n, 0).

G.f.: 3*x^6 -x^7*(x-2)*(2*x^4-11*x^3+24*x^2-25*x+11)/(x-1)^6. Simon Plouffe in his 1992 dissertation

a(n) = (n+4)*(n-3)*(n^3-6*n^2+3*n-10)/120, n >= 7. - R. J. Mathar, May 19 2013

Mathematica

Join[{3}, Table[Binomial[n, 5]+Binomial[n, 4]-Binomial[n, 3]+1, {n, 7, 50}]] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {3, 22, 71, 169, 343, 628, 1068}, 50] (* Harvey P. Dale, Aug 30 2021 *)

Crossrefs

Cf. A008302.

Sequence in context: A006532 A274870 A178492 * A143166 A268792 A055550

Adjacent sequences: A005285 A005286 A005287 * A005289 A005290 A005291

Keyword

easy,nonn

Author

N. J. A. Sloane

Status

approved