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A176280 Diagonal sums of number triangle A046521.

%I A176280 #19 Mar 30 2025 11:23:57
%S A176280 1,2,7,26,101,402,1625,6638,27319,113054,469811,1958706,8187063,
%T A176280 34290934,143864999,604402050,2542083509,10702020746,45090876913,
%U A176280 190110250998,801997354525,3384971428258,14292950533517,60373808435046,255102065046401,1078202260326002
%N A176280 Diagonal sums of number triangle A046521.
%C A176280 Hankel transform is A176281.
%H A176280 Vincenzo Librandi, <a href="/A176280/b176280.txt">Table of n, a(n) for n = 0..200</a>
%F A176280 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(2*(n-k),n-k)/C(2*k,k).
%F A176280 From _Vaclav Kotesovec_, Oct 21 2012: (Start)
%F A176280 G.f.: sqrt(1-4*x)/(1-4*x-x^2).
%F A176280 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 A176280 a(n) ~ (2+sqrt(5))^n/(2*sqrt(5)). (End)
%F A176280 a(n) = 4^n*binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4). - _Peter Luschny_, Mar 30 2025
%p A176280 seq(coeff(series(sqrt(1-4*x)/(1-4*x-x^2), x, n+1), x, n), n = 0..30);
%p A176280 # _G. C. Greubel_, Nov 24 2019
%p A176280 a := n -> 4^n*binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4):
%p A176280 seq(simplify(a(n)), n = 0..25); # _Peter Luschny_, Mar 30 2025
%t A176280 CoefficientList[Series[Sqrt[1-4*x]/(1-4*x-x^2), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Oct 21 2012 *)
%o A176280 (PARI) my(x='x+O('x^30)); Vec(sqrt(1-4*x)/(1-4*x-x^2)) \\ _G. C. Greubel_, Nov 24 2019
%o A176280 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)/(1-4*x-x^2) )); // _G. C. Greubel_, Nov 24 2019
%o A176280 (Sage)
%o A176280 def A176280_list(prec):
%o A176280     P.<x> = PowerSeriesRing(ZZ, prec)
%o A176280     return P( sqrt(1-4*x)/(1-4*x-x^2) ).list()
%o A176280 A176280_list(30) # _G. C. Greubel_, Nov 24 2019
%K A176280 nonn,easy
%O A176280 0,2
%A A176280 _Paul Barry_, Apr 14 2010