A084772 - OEIS

%I #64 May 22 2024 01:28:17

%S 1,6,46,396,3606,33876,324556,3151896,30915046,305543556,3038019876,

%T 30354866856,304523343996,3065412858696,30946859111256,

%U 313206733667376,3176825392214406,32284147284682596,328643023505612596

%N Coefficients of 1/sqrt(1 - 12*x + 16*x^2); also, a(n) is the central coefficient of (1 + 6*x + 5*x^2)^n.

%C Diagonal of rational functions 1/(1 - x - y - 4*x*y), 1/(1 - x - y*z - 4*x*y*z). - _Gheorghe Coserea_, Jul 06 2018

%H Vincenzo Librandi, <a href="/A084772/b084772.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 H. A. Verrill, <a href="http://arxiv.org/abs/math/0407327">Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations</a>, arXiv:math/0407327 [math.CO], 2008, Theorem 8.

%F a(n) = Sum_{k=0..n} 5^k*C(n,k)^2. - _Benoit Cloitre_, Oct 26 2003

%F E.g.f.: exp(6*x)*BesselI(0, 2*sqrt(5)*x). - _Paul Barry_, Sep 20 2004

%F Asymptotic: a(n) ~ (1+sqrt(5))^(2*n+1)/(2*5^(1/4)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Sep 11 2012

%F D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) - 16*(n-1)*a(n-2). - _R. J. Mathar_, Nov 09 2012

%F a(n) = 4^n * LegendreP(n, 3/2). - _G. C. Greubel_, May 31 2023

%F a(n) = 4^n*hypergeom([-n, n + 1], [1], -1/4). - _Detlef Meya_, May 22 2024

%e 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.

%t Table[n! SeriesCoefficient[E^(6 x) BesselI[0, 2 Sqrt[5] x], {x, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, May 10 2013 *)

%t CoefficientList[Series[1/Sqrt[1-12x+16x^2],{x,0,30}],x] (* _Harvey P. Dale_, Apr 17 2015 *)

%t Table[4^n*LegendreP[n,3/2], {n,0,40}] (* _G. C. Greubel_, May 31 2023 *)

%t a[n_] := 4^n*HypergeometricPFQ[{-n, n + 1}, {1}, -1/4]; Flatten[Table[a[n], {n,0,18}]] (* _Detlef Meya_, May 22 2024 *)

%o (PARI) for(n=0,30,t=polcoeff((1+6*x+5*x^2)^n,n,x); print1(t","))

%o (GAP) List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*5^k)); # _Muniru A Asiru_, Jul 29 2018

%o (Magma) [4^n*Evaluate(LegendrePolynomial(n), 3/2) : n in [0..40]]; // _G. C. Greubel_, May 31 2023

%o (SageMath) [4^n*gen_legendre_P(n, 0, 3/2) for n in range(41)] # _G. C. Greubel_, May 31 2023

%Y Cf. A001850.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 10 2003