A014533 Form array in which n-th row is obtained by expanding (1 + x + x^2)^n and taking the 4th column from the center.

1, 5, 21, 77, 266, 882, 2850, 9042, 28314, 87802, 270270, 827190, 2520336, 7651632, 23162976, 69954048, 210859245, 634569201, 1907165337, 5725520801, 17172595110, 51465297950, 154135675070, 461366154990, 1380317174145

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

First differences seem to be in A025182.

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

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Trinomial Coefficient

Programs

Maple

a := n -> simplify(GegenbauerC(n-1, -n-3, -1/2)):
seq(a(n), n=1..25); # Peter Luschny, May 09 2016

Mathematica

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 *)
Table[GegenbauerC[n-1, -n - 3, -1/2], {n,0,50}] (* G. C. Greubel, Feb 28 2017 *)

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

Formula

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

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

a(n) ~ 3^(n+7/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 20 2015

From Peter Luschny, May 09 2016: (Start)

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

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

Extensions

More terms from James Sellers, Feb 05 2000