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A144507 Column 4 of triangle in A144505.

%I A144507 #17 Oct 10 2023 05:02:21
%S A144507 0,0,0,0,0,1,20,330,5495,97405,1867446,38849790,875734035,21320230140,
%T A144507 558453090910,15677076200786,469894617088260,14985440023696415,
%U A144507 506831098757070010,18125347345533260190,683518670893880841921,27112243165544881804755,1128576366359460556636770
%N A144507 Column 4 of triangle in A144505.
%H A144507 G. C. Greubel, <a href="/A144507/b144507.txt">Table of n, a(n) for n = 0..400</a>
%F A144507 a(n) = (1/4!)*Sum_{k=0..n-5} (n+k-1)!/((n-k-5)!*k!*2^k).
%F A144507 a(n) = A001516(n-3)/6 for n > 2. [Corrected by _Georg Fischer_, Jan 25 2020]
%F A144507 a(n) = ( (2*n-7)*(n^2 -7*n +14)*a(n-1) + (n-2)*(n-3)*a(n-2) )/((n-4)*(n-5)), with a(0)=a(1)=a(2)=a(3)=a(4)=0, and a(5)=1. - _G. C. Greubel_, Oct 10 2023
%p A144507 f4:=proc(n) local k; add((n+k-1)!/(4!*(n-k-5)!*k!*2^k),k=0..n-5); end;
%p A144507 [seq(f4(n), n=0..60)];
%t A144507 Table[Sum[1/6 (n+k+2)!/(2^(k+2) (n-k-2)! k!), {k,0,n-2}], {n, -3, 20}] (* _Vincenzo Librandi_, Jan 27 2020 *)
%o A144507 (Magma) I:=[0,0,0,0,0,1]; [n le 6 select I[n] else ((2*n-9)*(n^2-9*n+22)*Self(n-1) + (n-3)*(n-4)*Self(n-2))/((n-5)*(n-6)): n in [1..32]]; // _G. C. Greubel_, Oct 10 2023
%o A144507 (SageMath)
%o A144507 @CachedFunction
%o A144507 def A144507(n): return sum(binomial(n-5,j)*rising_factorial(n-4,j+4)/(24*2^j) for j in range(n-4))
%o A144507 [A144507(n) for n in range(31)] # _G. C. Greubel_, Oct 10 2023
%Y A144507 Cf. A001516, A144505, A144506.
%K A144507 nonn
%O A144507 0,7
%A A144507 _N. J. A. Sloane_, Dec 14 2008