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Form array in which n-th row is obtained by expanding (1 + x + x^2)^n and taking the 4th column from the center.
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%I #36 Feb 28 2017 12:41:35

%S 1,5,21,77,266,882,2850,9042,28314,87802,270270,827190,2520336,

%T 7651632,23162976,69954048,210859245,634569201,1907165337,5725520801,

%U 17172595110,51465297950,154135675070,461366154990,1380317174145

%N Form array in which n-th row is obtained by expanding (1 + x + x^2)^n and taking the 4th column from the center.

%C First differences seem to be in A025182.

%C a(n-3) = A111808(n, n-4) for n > 3. - _Reinhard Zumkeller_, Aug 17 2005

%C a(n-4) = number of paths in the half-plane x >= 0, from (0,0) to (n,4), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=5, we have the 5 paths HUUUU, UHUUU, UUHUU, UUUHU, UUUUH. - _José Luis Ramírez Ramírez_, Apr 19 2015

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

%H G. C. Greubel, <a href="/A014533/b014533.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>

%F Conjecture: -(n+7)*(n-1)*a(n) + (n+3)*(2*n+5)*a(n-1) + 3*(n+3)*(n+2)*a(n-2) = 0. - _R. J. Mathar_, Feb 25 2015

%F G.f.: z*M(z)^4/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - _José Luis Ramírez Ramírez_, Apr 19 2015

%F a(n) ~ 3^(n+7/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 20 2015

%F From _Peter Luschny_, May 09 2016: (Start)

%F a(n) = C(6+2*n, n-1)*hypergeom([-n+1, -n-7], [-5/2-n], 1/4).

%F a(n) = GegenbauerC(n-1, -n-3, -1/2). (End)

%p a := n -> simplify(GegenbauerC(n-1, -n-3, -1/2)):

%p seq(a(n), n=1..25); # _Peter Luschny_, May 09 2016

%t Rest[CoefficientList[Series[x*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2))^4/(1-x-2*x^2*(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Apr 20 2015 *)

%t Table[GegenbauerC[n-1, -n - 3, -1/2], {n,0,50}] (* _G. C. Greubel_, Feb 28 2017 *)

%o (PARI) x='x + O('x^50); Vec(x*((1-x-sqrt(1-2*x-3*x^2))/(2*x^2))^4/(1-x-2*x^2*(1-x-sqrt(1-2*x-3*x^2))/(2*x^2))) \\ _G. C. Greubel_, Feb 28 2017

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Feb 05 2000