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%I #20 Jan 30 2020 21:29:15
%S 1,10,160,2800,51400,970000,18640000,362800000,7128700000,
%T 141103000000,2809273600000,56197096000000,1128614356000000,
%U 22741607080000000,459548117440000000,9309106936000000000,188980474087000000000
%N Expansion of 1/sqrt(1 - 20*x - 20*x^2).
%C Central coefficient of (1 + 10x + 30x^2)^n. Tenth binomial transform of 1/sqrt(1 - 120x^2). In general, 1/sqrt(1 - 4*r*x - 4*r*x^2) has e.g.f. exp(2rx)*BesselI(0,2r*sqrt((r+1)/r)x)), and a(n) = Sum_{k=0..n} C(2k,k)*C(k,n-k)*r^k gives the central coefficient of (1 + (2r)*x + r(r+1)*x^2) and is the (2r)-th binomial transform of 1/sqrt(1 - 8*C(n+1,2)x^2).
%H G. C. Greubel, <a href="/A106261/b106261.txt">Table of n, a(n) for n = 0..750</a>
%H Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%F E.g.f.: exp(10*x)*BesselI(0, 10*sqrt(6/5)*x).
%F a(n) = Sum_{k=0..n} C(2k, k)*C(k, n-k)*5^k.
%F D-finite with recurrence: n*a(n) + 10*(-2*n+1)*a(n-1) + 20*(-n+1)*a(n-2) = 0. - _R. J. Mathar_, Nov 26 2012
%F a(n) ~ sqrt((1+sqrt(5/6))/2) * (10+2*sqrt(30))^n / sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 19 2013
%t CoefficientList[Series[1/Sqrt[1-20*x-20*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2013 *)
%o (PARI) for(n=0,25, print1(sum(k=0,n,binomial(2*k,k)*binomial(k,n-k)*5^k), ", ")) \\ _G. C. Greubel_, Jan 31 2017
%Y Cf. A006139, A106258, A106259, A106260.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 28 2005
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