A000338 Expansion of x^3*(5-2*x)*(1-x^3)/(1-x)^4.

5, 18, 42, 75, 117, 168, 228, 297, 375, 462, 558, 663, 777, 900, 1032, 1173, 1323, 1482, 1650, 1827, 2013, 2208, 2412, 2625, 2847, 3078, 3318, 3567, 3825, 4092, 4368, 4653, 4947, 5250, 5562, 5883, 6213, 6552, 6900, 7257, 7623, 7998, 8382, 8775, 9177, 9588, 10008, 10437, 10875, 11322, 11778

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 3..1000
• 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
• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Maple

ff := n->9/2*n^2-15/2*n; seq(ff(n), n=3..60); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001, sequence without a(3).

Mathematica

nn = 100; CoefficientList[Series[(5 - 2 x) (1 - x^3)/(1 - x)^4, {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
LinearRecurrence[{3,-3,1},{5,18,42,75},60] (* Harvey P. Dale, Sep 20 2016 *)

Formula

a(n) = 3*A095794(n-2), n>3. - R. J. Mathar, May 30 2022

G.f.: (1+x+x^2)*(5-2*x)*x^3/(1-x)^3. - Simon Plouffe in his 1992 dissertation

Sum_{n>=3} 1/a(n) = log(3)/5 + Pi*sqrt(3)/45 = 0.3406424... - R. J. Mathar, Apr 22 2024

a(n) = 5*A005448(n-2) -2*A005448(n-3). - R. J. Mathar, Apr 22 2024

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001