A144506 Column 3 of triangle in A144505, negated.

0, 0, 0, 0, 1, 14, 175, 2330, 34300, 561386, 10179309, 203240850, 4439192835, 105413331100, 2705921548616, 74703337429084, 2207904948683525, 69575538504102190, 2329022305536291275, 82546355086989894366, 3088417981826529182964, 121651432581579519835950, 5032424258902838518567945

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

Programs

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

Maple

f3:=proc(n) local k; add((n+k-1)!/(6*(n-k-4)!*k!*2^k),k=0..n-4); end;
[seq(f3(n), n=0..60)];

Mathematica

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 *)
Table[a[n], {n,0,30}] (* G. C. Greubel, Oct 10 2023 *)

SageMath

@CachedFunction
def A144506(n): return sum(binomial(n-4,j)*rising_factorial(n-3,j+3)/(6*2^j) for j in range(n-3))
[A144506(n) for n in range(31)] # G. C. Greubel, Oct 10 2023

Formula

a(n) = Sum_{k=0..n-4} (n+k-1)!/(6*k!*(n-k-4)!*2^k).

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