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A006139 n*a(n) = 2*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2).
(Formerly M1849)
17
1, 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152, 11258054144, 52767312896, 247736643584, 1164829376512, 5484233814016, 25852072517632, 121997903495168 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = number of Delannoy paths (A001850) from (0,0) to (n,n) in which every Northeast step is immediately preceded by an East step. - David Callan, Mar 14 2004

The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005

In general, 1/sqrt(1-4*r*x-4*r*x^2) has e.g.f. exp(2rx)BesselI(0,2r*sqrt((r+1)/r)x), a(n)=sum{k=0..n, C(2k,k)C(k,n-k)r^k}, gives the central coefficient of (1+(2r)x+r(r+1)x^2) and is the (2r)-th binomial transform of 1/sqrt(1-8*C(n+1,2)x^2). - Paul Barry, Apr 28 2005

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H and U steps can have two colors. - N-E. Fahssi, Feb 05 2008

REFERENCES

D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

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

FORMULA

Sum(binomial(2*k, k)*binomial(k, n-k), k=0..n). - Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001

G.f.: 1/(1-4x-4x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+2x^2)^n. - Paul D. Hanna, Jun 01 2003

Inverse binomial transform of central Delannoy numbers A001850. - David Callan, Mar 14 2004

E.g.f.: exp(2*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004

a(n) = sum{k=0..floor(n/2), C(n,2k)C(2k,k)2^(n-k)}. - Paul Barry, Sep 19 2006

a(n) ~ (2+2*sqrt(2))^n/sqrt((4-2*sqrt(2))*Pi*n). - Vaclav Kotesovec, Oct 05 2012

G.f.: 1/(1 - 2*x*(1+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

a(n) = 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2). - Peter Luschny, Sep 18 2014

MAPLE

seq( sum('binomial(2*k, k)*binomial(k, n-k)', 'k'=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001

A006139 := n -> 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2);

seq(round(evalf(A006139(n), 99)), n=0..29); # Peter Luschny, Sep 18 2014

MATHEMATICA

Table[SeriesCoefficient[1/(1-4x-4x^2)^(1/2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 05 2012 *)

PROG

(PARI) for(n=0, 30, t=polcoeff((1+2*x+2*x^2)^n, n, x); print1(t", "))

CROSSREFS

Cf. A002426, A084600-A084606, A084608-A084615.

Cf. A106258, A106259, A106260, A106261.

First column of A110446.

Sequence in context: A150830 A150831 A084607 * A150832 A150833 A198760

Adjacent sequences:  A006136 A006137 A006138 * A006140 A006141 A006142

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 20 17:47 EST 2014. Contains 252272 sequences.