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A006139 n*a(n) = 2*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2).
(Formerly M1849)
18
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

Self-convolution of a(n)/2^n gives Pell numbers A000129(n+1). - Vladimir Reshetnikov, Oct 10 2016

REFERENCES

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

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

M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv:1410.5747 [math.CO], 2014.

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

0 = a(n)*(+16*a(n+1) + 24*a(n+2) - 8*a(n+3)) + a(n+1)*(+8*a(n+1) + 16*a(n+2) - 6*a(n+3)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Oct 13 2016

EXAMPLE

G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 136*x^4 + 592*x^5 + 2624*x^6 + 11776*x^7 + ...

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 *)

Table[Abs[LegendreP[n, I]] 2^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)

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 4 17:26 EST 2016. Contains 278755 sequences.