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A125558 Central column of triangle A090181. 5

%I

%S 1,1,6,50,490,5292,60984,736164,9202050,118195220,1551580888,

%T 20734762776,281248448936,3863302870000,53644719852000,

%U 751920156592200,10626401036545650,151269944167296900,2167317913508055000

%N Central column of triangle A090181.

%C [1,6,50,490,5292,...] is a column in triangle of Narayana numbers A001263.

%C Number of Dyck 2n-paths with exactly n peaks. - _Peter Luschny_, May 10 2014

%H Harvey P. Dale, <a href="/A125558/b125558.txt">Table of n, a(n) for n = 0..800</a>

%F a(0)=1, a(n) = Catalan(n)^2*(n+1)/2 = A000108(n)^2*(n+1)/2 for n>0.

%F a(n) = A090181(2*n, n).

%F G.f.: 1 + x*3F2( 1, 3/2, 3/2; 2, 3;16 x) = 1 + ( 2F1( 1/2, 1/2; 2;16*x) - 1)/2. - _Olivier Gérard_, Feb 16 2011

%F D-finite with recurrence n*(n+1)*a(n) -4*(2*n-1)^2*a(n-1)=0. - _R. J. Mathar_, Feb 08 2021

%p seq(ceil(1/2*(n+1)*((binomial(2*n,n)/(1+n))^2)), n=0..18); # _Zerinvary Lajos_, Jun 18 2007

%t CoefficientList[

%t Series[1 + (HypergeometricPFQ[{1/2, 1/2}, {2}, 16 x] - 1)/(2), {x, 0,

%t 20}], x]

%t Join[{1},Table[CatalanNumber[n]^2 (n+1)/2,{n,20}]] (* _Harvey P. Dale_, Oct 19 2011 *)

%Y Equals A000888(n)/2 for n>0.

%Y Cf. A090181.

%K easy,nonn

%O 0,3

%A _Philippe Deléham_, Jan 01 2007, Oct 11 2007

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Last modified February 28 20:48 EST 2021. Contains 341731 sequences. (Running on oeis4.)