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%I #34 Sep 11 2021 06:59:32
%S 1,7,61,595,6145,65527,712909,7863667,87615745,983726695,11112210781,
%T 126142119187,1437751935361,16443380994775,188609259215725,
%U 2168833084841395,24994269200292865,288596644195946695,3337978523215692925,38666734085509918675
%N Expansion of 1/sqrt((1-7*x)^2-24*x^2).
%H Vincenzo Librandi, <a href="/A098659/b098659.txt">Table of n, a(n) for n = 0..200</a>
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H Hacène Belbachir, Abdelghani Mehdaoui, and 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.
%H Rob Noble, <a href="https://doi.org/10.1016/0022-314X(84)90074-X">Asymptotics of a family of binomial sums</a>, J. Number Theory 130 (2010), no. 11, 2561-2585.
%F G.f.: 1/sqrt(1-14*x+25*x^2).
%F E.g.f.: exp(7*x)*BesselI(0, 2*sqrt(6)*x).
%F a(n) = Sum_{k=0..n} C(n, k)^2*6^k.
%F a(n) = [x^n] (1+7*x+6*x^2)^n.
%F From _Vaclav Kotesovec_, Sep 15 2012: (Start)
%F General recurrence for Sum_{k=0..n} C(n,k)^2*x^k (this is case x=6): (n+2)*a(n+2)-(x+1)*(2*n+3)*a(n+1)+(x-1)^2*(n+1)*a(n)=0.
%F Asymptotic (Rob Noble, 2010): a(n) ~ (1+sqrt(x))^(2*n+1)/(2*x^(1/4)*sqrt(Pi*n)), this is case x=6.
%F (End)
%F D-finite with recurrence: n*a(n) +7*(-2*n+1)*a(n-1) +25*(n-1)*a(n-2)=0. - _R. J. Mathar_, Jan 20 2020
%t Table[Sum[Binomial[n, k]^2*6^k, {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Sep 15 2012 *)
%t CoefficientList[Series[1/Sqrt[(1-7*x)^2-24*x^2], {x, 0, 25}], x] (* _Stefano Spezia_, Dec 04 2018 *)
%o (PARI) x='x+O('x^66); Vec(1/sqrt(1-14*x+25*x^2)) \\ _Joerg Arndt_, May 12 2013
%Y Cf. A000984, A001850, A069835, A084771, A084772.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 20 2004
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