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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 16 00:11 EST 2018. Contains 317252 sequences. (Running on oeis4.)