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Expansion of 1/((1-x)^4 - 8*x^4)^(1/4).
1

%I #15 Oct 08 2024 07:19:40

%S 1,1,1,1,3,11,31,71,151,343,871,2311,6001,15081,37493,94381,241931,

%T 625771,1617211,4164763,10719793,27674473,71722773,186353453,

%U 484657729,1260984161,3283294561,8559401761,22343836711,58391858383,152722920691,399719304411

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

%F a(n) = Sum_{k=0..floor(n/4)} (-8)^k * binomial(-1/4,k) * binomial(n,n-4*k).

%F (7 + 7*n)*a(n) + (7 + 4*n)*a(n + 1) - (15 + 6*n)*a(n + 2) + (13 + 4*n)*a(n + 3) - (n + 4)*a(n + 4) = 0. - _Robert Israel_, Oct 06 2024

%p f:= 1/((1-x)^4 - 8*x^4)^(1/4):

%p S:= series(f,x,41):

%p seq(coeff(S,x,i),i=0..40); # _Robert Israel_, Oct 06 2024

%t a[n_]:=Sum[(-8)^k * Binomial[-1/4,k] * Binomial[n,n-4*k],{k,0,Floor[n/4]}]; Array[a,32,0] (* _Stefano Spezia_, Oct 06 2024 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/((1-x)^4-8*x^4)^(1/4))

%Y Cf. A002426, A376836.

%Y Cf. A004981.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Oct 06 2024