A000574 Coefficient of x^5 in expansion of (1 + x + x^2)^n. (Formerly M3011 N1219)

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, 932178, 1053208

Offset

3,1

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

Louis 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, D. P. Roselle, and R. A. Scoville, 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).

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 5 < n else 2*n-5. - Peter 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

From Amiram Eldar, Oct 21 2025: (Start)

Sum_{n>=3} 1/a(n) = 31200607/72144072.

Sum_{n>=3} (-1)^(n+1)/a(n) = 1840*log(2)/143 - 622879391/72144072. (End)

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`(5<n, 5, 2*n-5), -n, -1/2):
seq(simplify(A000574(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 *)

Prog

(Magma) [3*Binomial(n+2, 5)-2*Binomial(n+1, 5): n in [3..50]]; // Vincenzo Librandi, Jun 10 2012
(PARI) my(x='x+O('x^50)); Vec(x^3*(3-2*x)/(1-x)^6) \\ G. C. Greubel, Nov 22 2017

Crossrefs

Cf. A000389, A005581, A005712, A005714, A005715, A005716, A111808.

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

Cf. A000217, A055998.

Sequence in context: A389329 A389377 A089363 * A041233 A055194 A190730

Adjacent sequences: A000571 A000572 A000573 * A000575 A000576 A000577

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Oct 02 2000

Status

approved