A027908 a(n) = T(2*n, n), T given by A027907.

1, 2, 10, 50, 266, 1452, 8074, 45474, 258570, 1481108, 8533660, 49402850, 287134346, 1674425300, 9792273690, 57407789550, 337281021450, 1985342102964, 11706001102180, 69124774458092, 408737856117916, 2419833655003752, 14341910428953018, 85087759173024870

Offset

0,2

Comments

Central terms of the triangle in A111808. - Reinhard Zumkeller, Aug 17 2005

Number of paths of semilength n starting at (0,0) and ending on the X-axis using steps (1,1), (1,-1) and (1,3). - David Scambler, Jun 21 2013

Links

Table of n, a(n) for n=0..23.

Formula

G.f.: -(g^2+g+1)/(3*g^2+g-1) where g = x times the g.f. of A143927. - Mark van Hoeij, Nov 16 2011

a(n) = GegenbauerC(n, -2*n, -1/2). - Peter Luschny, May 09 2016

From Peter Bala, Jan 26 2020: (Start)

a(n) = [x^(2*n)](1 + x^2 + x^4)^(2*n).

a(n) = Sum_{k = 0..floor(n/2)} C(2*n, n-k)*C(n-k, k).

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

Conjectural: a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for all primes p >= 5 and positive integers n and k. (End)

From Peter Bala, Aug 03 2023: (Start)

P-recursive: 3*n*(13*n - 17)*(3*n - 1)*(3*n - 2)*a(n) = 2*(2*n - 1)*(455*n^3 - 1050*n^2 + 691*n - 120)*a(n-1) + 36*(n - 1)*(13*n - 4)*(2*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.

exp(Sum_{n >= 0} a(n)*x^n/n) = 1 + 2*x + 7*x^2 + 28*x^3 + 123*x^4 + ... is the g.f. of A143927.

a(n) = 2*A344396(n-1) for n >= 1. (End)

Maple

ogf := series( RootOf( (144*x^2+140*x-27)*g^4+(18-12*x)*g^2+8*g+1, g), x=0, 20); # Mark van Hoeij, Nov 16 2011
a := n -> simplify(GegenbauerC(n, -2*n, -1/2)):
seq(a(n), n=0..23); # Peter Luschny, May 09 2016

Mathematica

Table[Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, 1/2 - 2 n, 1/4], {n, 0, 20}] (* or *) Table[GegenbauerC[3 n, -2 n, -1/2] + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 07 2016 *)

Prog

(Maxima) makelist(ultraspherical(n, -2*n, -1/2), n, 0, 12); /* Emanuele Munarini, Oct 18 2016 */

Crossrefs

Cf. A027907, A027913, A143927, A344396.

Sequence in context: A020729 A110170 A026332 * A206637 A318494 A020088

Adjacent sequences: A027905 A027906 A027907 * A027909 A027910 A027911

Keyword

nonn

Author

Clark Kimberling

Status

approved