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Expansion of (1 - 2*x + 2*x^2)/(1 - 2*x - 3*x^2)^(7/2).
3

%I #18 Aug 07 2024 22:43:36

%S 1,5,30,140,630,2646,10710,41910,159885,597025,2190188,7914270,

%T 28230020,99567300,347720040,1203777072,4135047615,14105322315,

%U 47813634330,161154659820,540353553894,1803226621350,5991410183850,19827295283250,65371101643575

%N Expansion of (1 - 2*x + 2*x^2)/(1 - 2*x - 3*x^2)^(7/2).

%F a(n) = (binomial(n+4,3)/4) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).

%F a(n) = (binomial(n+4,3)/4) * A005717(n+1).

%F a(n) = ((n+4)/(n*(n+2))) * ((2*n+1)*a(n-1) + 3*(n+3)*a(n-2)).

%F a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([(1-n)/2, -n/2], [2], 4)/24. - _Stefano Spezia_, Aug 07 2024

%t a[n_]:=(1+n)(2+n)(3+n)(4+n)Hypergeometric2F1[(1-n)/2,-n/2,2,4]/24; Array[a,25,0] (* _Stefano Spezia_, Aug 07 2024 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec((1-2*x+2*x^2)/(1-2*x-3*x^2)^(7/2))

%Y Column k=4 of A091869 (with a different offset).

%Y Cf. A374506, A375248.

%Y Cf. A005717, A373651.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 07 2024