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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

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

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

Programs

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

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`(4A005712(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 *)

PARI

Vec((x^2)*(1+x-x^2)/(1-x)^5+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

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 4Peter Luschny, May 10 2016

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

Extensions

More terms from Vladeta Jovovic, Oct 02 2000