A098443 - OEIS

%I #49 Mar 21 2024 07:08:09

%S 1,4,26,184,1366,10424,80996,637424,5064166,40528984,326251276,

%T 2638751504,21426682876,174563719984,1426219233416,11681133293024,

%U 95877105146246,788433553532824,6494463369141116,53576199709855184

%N Expansion of 1/sqrt(1-8*x-4*x^2).

%C Binomial transform of A098444. Second binomial transform of A084770. Third binomial transform of A098264.

%H Vincenzo Librandi, <a href="/A098443/b098443.txt">Table of n, a(n) for n = 0..200</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.

%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

%F E.g.f.: exp(4x)*BesselI(0, 2*sqrt(5)*x).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), n)*2^(n-2k).

%F D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012

%F a(n) ~ sqrt(50+20*sqrt(5))*(4+2*sqrt(5))^n/(10*sqrt(Pi*n)). Equivalently, a(n) ~ 2^(n-1/2) * phi^(3*n + 3/2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Oct 15 2012, updated Mar 21 2024

%F G.f.: 1/(1 - 2*x*(2+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(2+x)/(k+1 - x*(2+x)*(2*k+2)*(4*k+3)/(2*x*(2+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 15 2013

%F G.f.: Q(0), where Q(k) = 1 + 2*x*(x+2)*(4*k+1)/( 2*k+1 - x*(x+2)*(2*k+1)*(4*k+3)/(x*(x+2)*(4*k+3) + (k+1)/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 16 2013

%F From _Peter Bala_, March 16 2024: (Start)

%F a(n) = (-2*i)^n * P(n, 2*i), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial.

%F Sum_{n >= 1} (-1)^(n+1)*4^n/(n*a(n-1)*a(n)) = 2*arctan(1/2) = 2*A073000. (End)

%e G.f. = 1 + 4*x + 26*x^2 + 184*x^3 + 1366*x^4 + 10424*x^5 + 80996*x^6 + ...

%t CoefficientList[Series[1/Sqrt[1 - 8*x - 4*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 15 2012, updated Mar 21 2024 *)

%o (PARI) x='x+O('x^66); Vec(1/sqrt(1-8*x-4*x^2)) \\ _Joerg Arndt_, May 11 2013

%K easy,nonn

%O 0,2

%A _Paul Barry_, Sep 07 2004