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%I #30 Jul 29 2024 06:22:58
%S 1,3,13,61,295,1447,7151,35491,176597,880125,4390901,21920913,
%T 109486993,547018941,2733608905,13662695645,68294088535,341399727335,
%U 1706739347095,8532741458075,42660172763995,213287735579135,1066389745361635,5331765761680895
%N Row sums of triangle A046521.
%C Hankel transform is A082761. - _Paul Barry_, Apr 14 2010
%H G. C. Greubel, <a href="/A046748/b046748.txt">Table of n, a(n) for n = 0..1000</a>
%H T.-X. He, L. W. Shapiro, <a href="http://dx.doi.org/10.1016/j.laa.2017.06.025">Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group</a>, Lin. Alg. Applic. 532 (2017) 25-41, p 35.
%F a(n) = binomial(2*n, n)*Sum_{k=0..n} binomial(n, k)/binomial(2*k, k).
%F a(n) = 5^n - 2*A046714(n-1), A046714(-1) := 0.
%F a(n) = 5*a(n-1) - 2*A000108(n-1).
%F G.f.: sqrt(1-4*x)/(1-5*x).
%F a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 1, a(-1) := 0, a(0)=1 (homogeneous recursion).
%F a(n) = binomial(2*n,n)*hypergeom([ -n,1 ],[ 1/2 ],-1/4) (hypergeometric 2F1 form).
%F 0 = a(n)*(+400*a(n+1) - 330*a(n+2) + 50*a(n+3)) + a(n+1)*(-30*a(n+1) + 71*a(n+2) - 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - _Michael Somos_, May 25 2014
%F a(n) ~ 5^(n - 1/2). - _Vaclav Kotesovec_, Jul 07 2016
%F D-finite with recurrence n*a(n) +3*(-3*n+2)*a(n-1) +10*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Jul 23 2017
%e G.f. = 1 + 3*x + 13*x^2 + 61*x^3 + 295*x^4 + 1447*x^5 + 7151*x^6 + ...
%t a[ n_] := SeriesCoefficient[ Sqrt[ 1 - 4 x] / (1 - 5 x), {x, 0, n}]; (* _Michael Somos_, May 25 2014 *)
%t a[ n_] := Binomial[ 2 n, n] Hypergeometric2F1[ -n, 1, 1/2, -1/4]; (* _Michael Somos_, May 25 2014 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( 1 - 4*x + x * O(x^n)) / (1 - 5*x), n))}; /* _Michael Somos_, May 25 2014 */
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 40);
%o Coefficients(R!( Sqrt(1-4*x)/(1-5*x) )); // _G. C. Greubel_, Jul 28 2024
%o (SageMath)
%o def A046748_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( sqrt(1-4*x)/(1-5*x) ).list()
%o A046748_list(40) # _G. C. Greubel_, Jul 28 2024
%Y Cf. A000108, A046521, A046714, A082761.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_, Dec 11 1999
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