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%I #48 May 22 2024 01:25:10
%S 1,7,69,763,8881,106407,1298949,16065483,200630241,2524253767,
%T 31947470149,406281388443,5187375332881,66454791792487,
%U 853788052488069,10996378059281643,141934540736139201,1835494145265388167,23776671158743933509,308463567293772941883
%N Coefficients of 1/sqrt(1-14*x+9*x^2); also, a(n) is the central coefficient of (1+7x+10x^2)^n.
%C G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.
%C Diagonal of rational functions 1/(1 - x - 2*y - 3*x*y), 1/(1 - x - 2*y*z - 3*x*y*z). - _Gheorghe Coserea_, Jul 06 2018
%H Vincenzo Librandi, <a href="/A084774/b084774.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry1/barry242.html">On the Central Coefficients of Riordan Matrices</a>, Journal of Integer Sequences, 16 (2013), #13.5.1.
%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 Vaclav Kotesovec, <a href="http://www.kotesovec.cz/math_articles/kotesovec_binomial_asymptotics.pdf">Asymptotic of a sums of powers of binomial coefficients * x^k</a>, 2012.
%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * 2^k * 5^(n-k). - _Paul D. Hanna_, Sep 28 2012
%F D-finite with recurrence: n*a(n) = 7*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) ~ sqrt(200 + 70*sqrt(10))*(7 + 2*sqrt(10))^n/(20*sqrt(Pi*n)) = (sqrt(2) + sqrt(5))^(2*n+1)/(2*10^(1/4)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) = 3^n * LegendreP(n, 7/3). - _G. C. Greubel_, May 31 2023
%F a(n) = 3^n*hypergeom([-n, n + 1], [1], -2/3). - _Detlef Meya_, May 22 2024
%t Table[Sum[Binomial[n,k]^2*2^k*5^(n-k),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%t Table[n! SeriesCoefficient[E^(7 x) BesselI[0, 2 Sqrt[10] x], {x,0,n}], {n,0,20}] (* _Vincenzo Librandi_, May 10 2013 *)
%t Table[3^n*LegendreP[n, 7/3], {n,0,40}] (* _G. C. Greubel_, May 31 2023 *)
%t a[n_] := 3^n*HypergeometricPFQ[{-n, n + 1}, {1}, -2/3]; Flatten[Table[a[n], {n,0,19}]] (* _Detlef Meya_, May 22 2024 *)
%o (PARI) for(n=0,30,t=polcoeff((1+7*x+10*x^2)^n,n,x); print1(t","))
%o (PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*2^k*5^(n-k))} \\ _Paul D. Hanna_, Sep 28 2012
%o (GAP) List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*2^k*5^(n-k))); # _Muniru A Asiru_, Jul 29 2018
%o (Magma) [3^n*Evaluate(LegendrePolynomial(n), 7/3) : n in [0..40]]; // _G. C. Greubel_, May 31 2023
%o (SageMath) [3^n*gen_legendre_P(n, 0, 7/3) for n in range(41)] # _G. C. Greubel_, May 31 2023
%Y Cf. A006442, A084771.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 11 2003
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