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A069835 Define an array as follows: b(i,0)=b(0,j)=1, b(i,j)=2*b(i-1,j-1)+b(i-1,j)+b(i,j-1). Then a(n)=b(n,n). 12
1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544, 98365972, 700701808, 5011371964, 35961808432, 258805997752, 1867175631136, 13500088649734, 97794850668952, 709626281415076, 5157024231645616, 37528209137458516, 273431636191026064, 1994448720786816712 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2) and e.g.f. exp(2kx)BesselI(0,2sqrt(k^2-1)x). - Paul Barry, May 25 2005

Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where D can have two colors. - Paul Barry, May 25 2005

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have three colors and H steps can have four colors. - N-E. Fahssi, Mar 31 2008

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and two kinds of steps (1,1). - Joerg Arndt, Jul 01 2011

Hankel transform is 2^n*3^C(n+1,2)=(-1)^C(n+1,2)*A127946(n). - Paul Barry, Jan 24 2011

Central terms of triangle A152842. - Reinhard Zumkeller, May 01 2014

LINKS

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..250

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - N. J. A. Sloane, Oct 08 2012

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

FORMULA

a(n) = 2^n*LegendreP(n, 2) = 2^n*hypergeom([ -n, n+1], [1], -1/2) = 2^n*GegenbauerC(n, 1/2, 2) = Sum_{k=0..n} 3^k*binomial(n, k)^2. Recurrence: a(n) = 4*(2*n-1)/n*a(n-1)-4*(n-1)/n*a(n-2). G.f.: 1/sqrt(1-8*x+4*x^2). - Vladeta Jovovic, May 13 2003

a(n) = central coefficient of (1+4*x+3*x^2)^n. - Paul D. Hanna, Jun 03 2003

E.g.f.: exp(4*x)*Bessel_I(0, 2*sqrt(3)*x). - Paul Barry, Sep 20 2004

a(n) = sum(k=0..floor(n/2), C(n, k)*C(2*(n-k), n)*(-1)^k*2^(n-2*k) ). - Paul Barry, May 25 2005

a(n) = sum(k=0..n, C(n, k)*C(n+k, k)*2^(n-k) ). - Paul Barry, May 25 2005

a(n) = sum(k=0..n, C(n, k)^2*3^k ). - Paul Barry, Oct 15 2005

G.f.: 1/(1-4x-6x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-... (continued fraction). - Paul Barry Jan 24 2011

Asymptotic: a(n) ~ sqrt(1/2+1/sqrt(3))*(1+sqrt(3))^(2*n)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 11 2012

EXAMPLE

The array b is a rewriting of A081577:

1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,..

1,  4,  7, 10, 13, 16, 19, 22, 25, 28, 31,...

1,  7, 22, 46, 79,121,172,232,301,379,466,...

1, 10, 46,136,307,586,1000,1576,2341,3322,4546,...

1, 13, 79,307,886,2086,4258,7834,13327,21331,32521,...

MATHEMATICA

Table[Hypergeometric2F1[-n, -n, 1, 3], {n, 0, 21}] (* Arkadiusz Wesolowski, Aug 13 2012 *)

PROG

(PARI) a(n)=sum(k=0, n, binomial(n, k)^2*3^k)

(PARI) {a(n)=if(n<0, 0, polcoeff((1+4*x+3*x^2)^n, n))}

(PARI) /* as lattice paths: same as in A092566 but use */

steps=[[1, 0], [0, 1], [1, 1], [1, 1]];  /* note the double [1, 1] */

\\ Joerg Arndt, Jul 01 2011

(Haskell)

a069835 n = a081577 (2 * n) n  -- Reinhard Zumkeller, Mar 16 2014

CROSSREFS

Cf. A001850.

Sequence in context: A183281 A067120 A143648 * A007196 A091638 A142984

Adjacent sequences:  A069832 A069833 A069834 * A069836 A069837 A069838

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, May 03 2002

STATUS

approved

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Last modified December 4 21:33 EST 2016. Contains 278755 sequences.