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Expansion of 1/(1 - 4*x - 4*x^2)^(5/2).
3

%I #17 Oct 19 2024 08:31:23

%S 1,10,80,560,3640,22512,134400,781440,4451040,24939200,137865728,

%T 753625600,4080643840,21916106240,116877312000,619457482752,

%U 3265293719040,17128725519360,89462514606080,465434423336960,2412895587536896,12468681310412800,64242981906022400

%N Expansion of 1/(1 - 4*x - 4*x^2)^(5/2).

%F a(0) = 1, a(1) = 10; a(n) = (2*(2*n+3)*a(n-1) + 4*(n+3)*a(n-2))/n.

%F a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).

%F a(n) = 2^(n-3)*Pochhammer(n+1, 4)*hypergeom([(1-n)/2, -n/2], [3], 2)/3. - _Stefano Spezia_, Jul 10 2024

%F a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(k,n-k). - _Seiichi Manyama_, Oct 19 2024

%t a[n_]:=2^(n-3) Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, 2]/3; Array[a,23,0] (* _Stefano Spezia_, Jul 10 2024 *)

%o (PARI) a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));

%Y Cf. A006139, A374497, A374513.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 09 2024