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A026585 a(n)=T(n,n), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0. 3
1, 0, 2, 2, 8, 14, 40, 86, 222, 518, 1296, 3130, 7770, 19066, 47324, 117094, 291260, 724302, 1806220, 4507230, 11266718, 28188070, 70609316, 177023466, 444231564, 1115639586, 2803975860, 7052132546, 17748069294, 44693162266 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The signed sequence 1,0,2,-2,8,-14,... is the inverse binomial transform of A026569. - Paul Barry, Sep 09 2004

Hankel transform of a(n) is 2^n. Hankel transform of a(n+1) is 0,-4,0,16,0,-64,0,256,0,... or -2^(n+1)*[x^n](x/(1+x^2)). Hankel transform of a(n+2) is 2^(n+1)*A109613(n+1). [Paul Barry, Mar 23 2011]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

FORMULA

G.f.: sqrt[(1-x)/(1-x-4x^2)]. - Ralf Stephan, Jan 08 2004

From Paul Barry, Jul 01 2009: (Start)

G.f.: 1/(1-2x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-... (continued fraction);

a(0) = 1, a(n) = sum{k=0..floor(n/2), (k/(n-k))C(n-k,k)*A000984(k)}. (End)

a(n) = sum{k=0..floor(n/2), C(n-k-1,n-2k)*A000984(k)} = sum{k=0..floor(n/2), C(n-k-1,n-2k)*C(2k,k)}. [Paul Barry, Mar 23 2011]

Conjecture: n*a(n) +2*(-n+1)*a(n-1) +(-3*n+2)*a(n-2) +2*(2*n-5)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ (sqrt(17)+1)^(n-1/2) / (17^(1/4) * sqrt(Pi*n) * 2^(n-3/2)). - Vaclav Kotesovec, Feb 12 2014

MATHEMATICA

CoefficientList[Series[Sqrt[(1-x)/(1-x-4*x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

CROSSREFS

Sequence in context: A280399 A005633 A228661 * A229730 A248097 A098273

Adjacent sequences:  A026582 A026583 A026584 * A026586 A026587 A026588

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified June 25 22:15 EDT 2017. Contains 288730 sequences.