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A081207 Main diagonal of number square A081206. 2
1, 2, 3, 7, 16, 37, 89, 216, 529, 1307, 3248, 8111, 20339, 51176, 129143, 326717, 828374, 2104361, 5354979, 13647682, 34830191, 89000157, 227674188, 583017657, 1494365341, 3833592212, 9842373849, 25287895051, 65016153154, 167264946727 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = Sum_{k=0..n} (binomial(floor((n+k)/2), k))^2.

G.f.: (1+x)/sqrt(1-2x-x^2-2x^3+x^4) - Paul Barry, Jun 04 2005

Conjecture: n*(n-2)*a(n) +(-2*n^2+5*n-1)*a(n-1) +(-n^2+3*n-4)*a(n-2) +(-2*n^2+7*n-4)*a(n-3) +(n-1)*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 12 2012

a(n) ~ (5-sqrt(5)) * ((3+sqrt(5))/2)^n / (2*sqrt(14*sqrt(5)-30) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014

MATHEMATICA

Table[Sum[Binomial[Floor[(n+k)/2], k]^2, {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Oct 02 2011 *)

CoefficientList[Series[(1+x)/Sqrt[1-2x-x^2-2x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

PROG

(PARI) for(n=0, 25, print1(sum(k=0, n, (binomial(floor((n+k)/2), k))^2), ", ")) \\ G. C. Greubel, Feb 16 2017

(PARI) x='x+O('x^25); Vec((1+x)/sqrt(1-2*x-x^2-2*x^3+x^4)) \\ G. C. Greubel, Feb 16 2017

CROSSREFS

Cf. A051286, A081206.

Sequence in context: A289844 A153056 A235112 * A027118 A114582 A107387

Adjacent sequences:  A081204 A081205 A081206 * A081208 A081209 A081210

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 11 2003

STATUS

approved

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Last modified March 24 12:14 EDT 2019. Contains 321448 sequences. (Running on oeis4.)