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
Examples
First 3 rows from A249057: 1 4 1 5 4 1, so that a(0) = 1, a(1) = 4, a(2) = 5.
Links
- Clark Kimberling, Table of n, a(n) for n = 0..100
Programs
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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 *) -
PARI
f(n) = if (n, x + (n + 3)/f(n-1), 1); a(n) = polcoef(numerator(f(n)), 0); \\ Michel Marcus, Nov 25 2022
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