A176280 - OEIS

%I #16 Sep 08 2022 08:45:52

%S 1,2,7,26,101,402,1625,6638,27319,113054,469811,1958706,8187063,

%T 34290934,143864999,604402050,2542083509,10702020746,45090876913,

%U 190110250998,801997354525,3384971428258,14292950533517,60373808435046,255102065046401,1078202260326002

%N Diagonal sums of number triangle A046521.

%C Hankel transform is A176281.

%H Vincenzo Librandi, <a href="/A176280/b176280.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(2*(n-k),n-k)/C(2*k,k).

%F From _Vaclav Kotesovec_, Oct 21 2012: (Start)

%F G.f.: sqrt(1-4*x)/(1-4*x-x^2).

%F Recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3).

%F a(n) ~ (2+sqrt(5))^n/(2*sqrt(5)). (End)

%p seq(coeff(series(sqrt(1-4*x)/(1-4*x-x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Nov 24 2019

%t CoefficientList[Series[Sqrt[1-4*x]/(1-4*x-x^2), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Oct 21 2012 *)

%o (PARI) my(x='x+O('x^30)); Vec(sqrt(1-4*x)/(1-4*x-x^2)) \\ _G. C. Greubel_, Nov 24 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)/(1-4*x-x^2) )); // _G. C. Greubel_, Nov 24 2019

%o (Sage)

%o def A176280_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( sqrt(1-4*x)/(1-4*x-x^2) ).list()

%o A176280_list(30) # _G. C. Greubel_, Nov 24 2019

%K nonn,easy

%O 0,2

%A _Paul Barry_, Apr 14 2010