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

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

Author

N. J. A. Sloane

Keywords

• easy
• nonn

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

• 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).

Crossrefs

Cf. A008302.

Programs

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 *)

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