A108490 - OEIS

%I #18 Sep 10 2021 18:27:36

%S 1,2,10,56,268,1448,7864,42752,236368,1313696,7339552,41217920,

%T 232321984,1313731712,7449834880,42347380736,241225384192,

%U 1376662561280,7869527190016,45051709749248,258256281078784,1482218578159616

%N Expansion of 1/sqrt(1-4x-8x^2-24x^3+36x^4).

%C In general, Sum_{k=0..n} C(n-k,k)^2*a^k*b^(n-k) has expansion 1/sqrt(1-2bx-(2ab-b^2)x^2-2a*b^2*x^3+(ab)^2*x^4).

%H Robert Israel, <a href="/A108490/b108490.txt">Table of n, a(n) for n = 0..1301</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.

%F a(n) = Sum_{k=0..n} C(n-k, k)^2*3^k*2^(n-k).

%F D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +8*(-n+1)*a(n-2) +12*(-2*n+3)*a(n-3) +36*(n-2)*a(n-4)=0. - _R. J. Mathar_, Feb 20 2015

%F Recurrence confirmed using d.e. (72*x^3 - 36*x^2 - 8*x - 2)*g(x) + (36*x^4 - 24*x^3 - 8*x^2 - 4*x + 1)*g'(x) = 0 satisfied by the g.f.. - _Robert Israel_, Jan 07 2019

%p f:= gfun:-rectoproc({(72 + 36*n)*a(n) + (-60 - 24*n)*a(n + 1) + (-8*n - 24)*a(n + 2) + (-14 - 4*n)*a(n + 3) + (n + 4)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 10, a(3) = 56}, a(n), remember):

%p map(f, [$0..50]); # _Robert Israel_, Jan 07 2019

%t CoefficientList[Series[1/Sqrt[1-4x-8x^2-24x^3+36x^4], {x,0,30}], x]  (* _Harvey P. Dale_, Mar 13 2011 *)

%Y Cf. A108486.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jun 04 2005