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

Author

Clark Kimberling

Keywords

• nonn

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

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A027907, A027913, A143927, A344396, A370159, A370160.

Programs

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 *)

Maxima

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

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)