A000574 Coefficient of x^5 in expansion of (1 + x + x^2)^n.

3, 16, 51, 126, 266, 504, 882, 1452, 2277, 3432, 5005, 7098, 9828, 13328, 17748, 23256, 30039, 38304, 48279, 60214, 74382, 91080, 110630, 133380, 159705, 190008, 224721, 264306, 309256, 360096, 417384, 481712, 553707, 634032, 723387, 822510

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If Y is a 3-subset of an n-set X then, for n>=7, a(n-4) is the number of 5-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
• 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

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
• 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
• Eric Weisstein's World of Mathematics, Trinomial Coefficient
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000389, A005581, A005712, A005714-A005716, A111808.

Column m=5 of (1, 3) Pascal triangle A095660.

Cf. A000217, A055998.

Programs

Magma

[3*Binomial(n+2,5)-2*Binomial(n+1,5): n in [3..50]]; // Vincenzo Librandi, Jun 10 2012

Maple

A000574:=-(-3+2*z)/(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation
seq(3*binomial(n+2,5)-2*binomial(n+1,5),n=3..100); # Robert Israel, Aug 04 2015
A000574 := n -> GegenbauerC(`if`(5A000574(n)), n=3..20); # Peter Luschny, May 10 2016

Mathematica

CoefficientList[Series[(3-2*x)/(1-x)^6,{x,0,40}],x] (* Vincenzo Librandi, Jun 10 2012 *)

PARI

x='x+O('x^50); Vec(x^3*(3-2*x)/(1-x)^6) \\ G. C. Greubel, Nov 22 2017

Formula

G.f.: x^3*(3-2*x)/(1-x)^6.

a(n) = 3*binomial(n+2,5) - 2*binomial(n+1,5).

a(n) = A111808(n,5) for n>4. - Reinhard Zumkeller, Aug 17 2005

a(n) = binomial(n+1, 4)*(n+12)/5 = 3*b(n-3)-2*b(n-4), with b(n)=binomial(n+5, 5); cf. A000389.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Vincenzo Librandi, Jun 10 2012

a(n) = 3*binomial(n, 3) + 4*binomial(n, 4) + binomial(n, 5). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

a(n) = GegenbauerC(N, -n, -1/2) where N = 5 if 5Peter Luschny, May 10 2016

a(n) = Sum_{i=1..n-1} A000217(i)*A055998(n-1-i). - Bruno Berselli, Mar 05 2018

E.g.f.: exp(x)*x^3*(60 + 20*x + x^2)/120. - Stefano Spezia, Jul 09 2023

Extensions

More terms from Vladeta Jovovic, Oct 02 2000