A005712 Coefficient of x^4 in expansion of (1+x+x^2)^n. (Formerly M4129)

1, 6, 19, 45, 90, 161, 266, 414, 615, 880, 1221, 1651, 2184, 2835, 3620, 4556, 5661, 6954, 8455, 10185, 12166, 14421, 16974, 19850, 23075, 26676, 30681, 35119, 40020, 45415, 51336, 57816, 64889, 72590, 80955, 90021, 99826, 110409, 121810, 134070

Offset

2,2

Comments

a(n) = A111808(n,4) for n>3. - Reinhard Zumkeller, Aug 17 2005

If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-3) is the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007

Antidiagonal sums of the convolution array A213781. [Clark Kimberling, Jun 22 2012]

References

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

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 = 2..1000

Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.

R. K. Guy, Letter to N. J. A. Sloane, 1987

Milan Janjic, Two Enumerative Functions

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 (5,-10,10,-5,1).

Formula

G.f.: (x^2)*(1+x-x^2)/(1-x)^5.

a(n) = binomial(n+2,n-2) + binomial(n+1,n-2) - binomial(n,n-2). - Zerinvary Lajos, May 16 2006

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). Vincenzo Librandi, Jun 16 2012

a(n) = binomial(n,2) + 3*binomial(n,3) + binomial(n,4) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

a(n) = GegenbauerC(N, -n, -1/2) where N = 4 if 4<n else 2*n-4. - Peter Luschny, May 10 2016

E.g.f.: exp(x)*x^2*(12 + 12*x + x^2)/24. - Stefano Spezia, Jul 09 2023

Maple

seq(binomial(n+2, n-2) + binomial(n+1, n-2) - binomial(n, n-2), n=2..50); # Zerinvary Lajos, May 16 2006
A005712:=(-1-z+z**2)/(z-1)**5; # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.
A005712 := n -> GegenbauerC(`if`(4<n, 4, 2*n-4), -n, -1/2):
seq(simplify(A005712(n)), n=2..20); # Peter Luschny, May 10 2016

Mathematica

CoefficientList[Series[(1+x-x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 16 2012 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 6, 19, 45, 90}, 40] (* Harvey P. Dale, Apr 30 2015 *)

Prog

(Magma) I:=[1, 6, 19, 45, 90]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; Vincenzo Librandi, Jun 16 2012
(PARI) Vec((x^2)*(1+x-x^2)/(1-x)^5+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Crossrefs

Cf. A000574, A005581, A005714-A005716, A026729, A111808.

a(n)= A027907(n, 4), n >= 2 (fifth column of trinomial coefficients).

Sequence in context: A266938 A362602 A299265 * A299278 A298741 A070893

Adjacent sequences: A005709 A005710 A005711 * A005713 A005714 A005715

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Oct 02 2000

Status

approved