A144507 Column 4 of triangle in A144505.

0, 0, 0, 0, 0, 1, 20, 330, 5495, 97405, 1867446, 38849790, 875734035, 21320230140, 558453090910, 15677076200786, 469894617088260, 14985440023696415, 506831098757070010, 18125347345533260190, 683518670893880841921, 27112243165544881804755, 1128576366359460556636770

Offset: 0

Author

N. J. A. Sloane, Dec 14 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A001516, A144505, A144506.

Programs

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

Maple

f4:=proc(n) local k; add((n+k-1)!/(4!*(n-k-5)!*k!*2^k),k=0..n-5); end;
[seq(f4(n), n=0..60)];

Mathematica

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 *)

SageMath

@CachedFunction
def A144507(n): return sum(binomial(n-5,j)*rising_factorial(n-4,j+4)/(24*2^j) for j in range(n-4))
[A144507(n) for n in range(31)] # G. C. Greubel, Oct 10 2023

Formula

a(n) = (1/4!)*Sum_{k=0..n-5} (n+k-1)!/((n-k-5)!*k!*2^k).

a(n) = A001516(n-3)/6 for n > 2. [Corrected by Georg Fischer, Jan 25 2020]

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