A249060 Column 1 of the triangular array at A249057.

1, 4, 5, 24, 35, 192, 315, 1920, 3465, 23040, 45045, 322560, 675675, 5160960, 11486475, 92897280, 218243025, 1857945600, 4583103525, 40874803200, 105411381075, 980995276800, 2635284526875, 25505877196800, 71152682225625, 714164561510400, 2063427784543125

Offset

0,2

Links

Clark Kimberling, Table of n, a(n) for n = 0..100

Formula

From Derek Orr, Oct 21 2014: (Start)

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

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

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)

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

D-finite with recurrence: a(n) - (3+n)*a(n-2) = 0. - Georg Fischer, Nov 25 2022

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

Example

First 3 rows from A249057:
1
4    1
5    4    1,
so that a(0) = 1, a(1) = 4, a(2) = 5.

Mathematica

z = 30; p[x_, n_] := x + (n + 2)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}];
u = Numerator[t]; v1 = Flatten[CoefficientList[u, x]]; (* A249057 *)
v2 = u /. x -> 1  (* A249059 *)
v3 = u /. x -> 0  (* A249060 *)

Prog

(PARI) f(n) = if (n, x + (n + 3)/f(n-1), 1);
a(n) = polcoef(numerator(f(n)), 0); \\ Michel Marcus, Nov 25 2022

Crossrefs

Cf. A178647, A249057, A309491.

Sequence in context: A042123 A041531 A089499 * A042601 A219515 A248246

Adjacent sequences: A249057 A249058 A249059 * A249061 A249062 A249063

Keyword

nonn,easy

Author

Clark Kimberling, Oct 20 2014

Status

approved