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

1,2

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

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)

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 *)

Prog

(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

Crossrefs

Sequence in context: A245034 A126645 A026329 * A255453 A134770 A272787

Adjacent sequences: A014530 A014531 A014532 * A014534 A014535 A014536

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 05 2000

Status

approved