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A005712
Coefficient of x^4 in expansion of (1+x+x^2)^n.
(Formerly M4129)
30
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
Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.
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
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
a(n)= A027907(n, 4), n >= 2 (fifth column of trinomial coefficients).
Sequence in context: A266938 A362602 A299265 * A299278 A298741 A070893
KEYWORD
nonn,easy
EXTENSIONS
More terms from Vladeta Jovovic, Oct 02 2000
STATUS
approved