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A249060
Column 1 of the triangular array at A249057.
3
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
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 20 2014
STATUS
approved