A005288 - OEIS

%I M3090 #44 Apr 13 2022 13:25:17

%S 3,22,71,169,343,628,1068,1717,2640,3914,5629,7889,10813,14536,19210,

%T 25005,32110,40734,51107,63481,78131,95356,115480,138853,165852,

%U 196882,232377,272801,318649,370448,428758,494173,567322

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

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

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

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

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

%H F. N. David, M. G. Kendall and D. E. Barton, <a href="/A005288/a005288.pdf">Symmetric Function and Allied Tables</a>, Cambridge, 1966, p. 241-242. (Annotated scanned copy)

%H R. K. Guy, <a href="/A000707/a000707_2.pdf">Letter to N. J. A. Sloane with attachment, Mar 1988</a>

%H R. H. Moritz and R. C. Williams, <a href="http://www.jstor.org/stable/2690326">A coin-tossing problem and some related combinatorics</a>, Math. Mag., 61 (1988), 24-29.

%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

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

%F 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

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

%t 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 *)

%Y Cf. A008302.

%K easy,nonn

%O 6,1

%A _N. J. A. Sloane_