A249060 - OEIS

%I #24 Dec 10 2022 10:46:45

%S 1,4,5,24,35,192,315,1920,3465,23040,45045,322560,675675,5160960,

%T 11486475,92897280,218243025,1857945600,4583103525,40874803200,

%U 105411381075,980995276800,2635284526875,25505877196800,71152682225625,714164561510400,2063427784543125

%N Column 1 of the triangular array at A249057.

%H Clark Kimberling, <a href="/A249060/b249060.txt">Table of n, a(n) for n = 0..100</a>

%F From _Derek Orr_, Oct 21 2014: (Start)

%F a(2*n) = (2*n+3)*(2*n+1)!!/3, for n > 0.

%F a(2*n+1) = (n+2)!*2^(n+1), for n > 0.

%F For n > 2, if n is even, a(n)/[(n+1)*(n-1)*(n-3)*...*7*5] = n + 3 and if n is odd, a(n)/[(n+1)*(n-1)*(n-3)*...*6*4] = n + 3. (End)

%F a(n) = gcd_2((n+3)!,(n+3)!!), where gcd_2(b,c) denotes the second-largest common divisor of non-coprime integers b and c, as defined in A309491. - _Lechoslaw Ratajczak_, Apr 15 2021

%F D-finite with recurrence: a(n) - (3+n)*a(n-2) = 0. - _Georg Fischer_, Nov 25 2022

%F Sum_{n>=0} 1/a(n) = 3*sqrt(e*Pi/2)*erf(1/sqrt(2)) + 2*sqrt(e) - 6, where erf is the error function. - _Amiram Eldar_, Dec 10 2022

%e First 3 rows from A249057:

%e 1

%e 4    1

%e 5    4    1,

%e so that a(0) = 1, a(1) = 4, a(2) = 5.

%t z = 30; p[x_, n_] := x + (n + 2)/p[x, n - 1]; p[x_, 1] = 1;

%t t = Table[Factor[p[x, n]], {n, 1, z}];

%t u = Numerator[t]; v1 = Flatten[CoefficientList[u, x]]; (* A249057 *)

%t v2 = u /. x -> 1  (* A249059 *)

%t v3 = u /. x -> 0  (* A249060 *)

%o (PARI) f(n) = if (n, x + (n + 3)/f(n-1), 1);

%o a(n) = polcoef(numerator(f(n)), 0); \\ _Michel Marcus_, Nov 25 2022

%Y Cf. A178647, A249057, A309491.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Oct 20 2014