A361790 - OEIS

%I #20 Jul 11 2024 14:43:46

%S 1,2,-2,-8,6,42,-8,-228,-90,1210,1238,-6116,-10864,28574,80932,

%T -116248,-548010,339678,3455686,173208,-20452674,-14036418,113365140,

%U 156407916,-580805472,-1312098918,2659610562,9621079540,-9902139124,-64566648122,18521111032

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

%H Winston de Greef, <a href="/A361790/b361790.txt">Table of n, a(n) for n = 0..2497</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+3*k-1,n-k).

%F n*a(n) = -( (n-3)*a(n-1) + (6*n-6)*a(n-2) + 10*(n-3)*a(n-3) + 5*(n-4)*a(n-4) + (n-5)*a(n-5) ) for n > 4.

%F a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+2-k,3) * a(k).

%F a(n) = (-1)^(n+1)*Pochhammer(n,3)*hypergeom([1-n, 1+n/3, (4+n)/3, (5+n)/3], [5/4, 7/4, 2], 3^3/2^6)/3 for n > 0. - _Stefano Spezia_, Jul 11 2024

%t a[n_]:=(-1)^(n+1)Pochhammer[n,3]HypergeometricPFQ[{1-n,1+n/3,(4+n)/3, (5+n)/3}, {5/4,7/4,2}, 3^3/2^6]/3; Join[{1},Array[a,30]] (* _Stefano Spezia_, Jul 11 2024 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^4))

%o (PARI) a(n)=sum(k=0, n, (-1)^(n-k) * binomial(2*k,k) * binomial(n+3*k-1,n-k)) \\ _Winston de Greef_, Mar 24 2023

%Y Cf. A006139, A137635, A360133, A361791, A361792.

%K sign

%O 0,2

%A _Seiichi Manyama_, Mar 24 2023