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A144506 Column 3 of triangle in A144505, negated.

%I A144506 #12 Oct 10 2023 05:01:58
%S A144506 0,0,0,0,1,14,175,2330,34300,561386,10179309,203240850,4439192835,
%T A144506 105413331100,2705921548616,74703337429084,2207904948683525,
%U A144506 69575538504102190,2329022305536291275,82546355086989894366,3088417981826529182964,121651432581579519835950,5032424258902838518567945
%N A144506 Column 3 of triangle in A144505, negated.
%H A144506 G. C. Greubel, <a href="/A144506/b144506.txt">Table of n, a(n) for n = 0..400</a>
%F A144506 a(n) = Sum_{k=0..n-4} (n+k-1)!/(6*k!*(n-k-4)!*2^k).
%F A144506 a(n) = ( (n-3)*(4*n^2 - 24*n + 41)*a(n-1) + (n-2)*(2*n-5)*a(n-2) )/((n-4)*(2*n-7)), with a(0)=a(1)=a(2)=a(3)= 0, and a(4) = 1. - _G. C. Greubel_, Oct 10 2023
%p A144506 f3:=proc(n) local k; add((n+k-1)!/(6*(n-k-4)!*k!*2^k),k=0..n-4); end;
%p A144506 [seq(f3(n), n=0..60)];
%t A144506 a[n_]:= a[n]= If[n<4, 0, If[n==4, 1, ((n-3)*(4*n^2-24*n+41)*a[n-1] + (n -2)*(2*n-5)*a[n-2])/((n-4)*(2*n-7))]]; (* a = A144506 *)
%t A144506 Table[a[n], {n,0,30}] (* _G. C. Greubel_, Oct 10 2023 *)
%o A144506 (Magma) I:=[0,0,0,0,1]; [n le 5 select I[n] else ((n-4)*(4*n^2-32*n+69)*Self(n-1) + (n-3)*(2*n-7)*Self(n-2))/((n-5)*(2*n-9)): n in [1..30]]; // A144506 // _G. C. Greubel_, Oct 10 2023
%o A144506 (SageMath)
%o A144506 @CachedFunction
%o A144506 def A144506(n): return sum(binomial(n-4,j)*rising_factorial(n-3,j+3)/(6*2^j) for j in range(n-3))
%o A144506 [A144506(n) for n in range(31)] # _G. C. Greubel_, Oct 10 2023
%K A144506 nonn
%O A144506 0,6
%A A144506 _N. J. A. Sloane_, Dec 14 2008