OFFSET
1,2
COMMENTS
Define recursively the rational fractions R_n by: R_0(x)=1; R_{n+1}(x) = (R_n(x)*x/(1-x^2))'. 2*a(n) is the maximal integer that can be factored out of the numerator of R'_n -- staying with polynomials with integer coefficients.
Empirical observations: the prime factorizations of the a(n) follow a pattern: the 2-adic valuation of a(n) is the 2-adic valuation of n; the 3-adic valuation of a(n) is (n mod 2); for p a prime >= 5, the p-adic valuation of a(n) is 0 (if p-1 does not divide n), 1 (if p-1 divides n but p does not) or 2 (if both p-1 and p divide n). So, a(n) = 1 when n is odd, and the prime factorizations of a(n) for the first few even n are:
\ p|
\ | 2 3 5 7 11 13 17 19 23 29 31 37 41 43
n \|
---+-------------------------------------------
2 | 1 1 . . . . . . . . . . . .
4 | 2 1 1 . . . . . . . . . . .
6 | 1 1 . 1 . . . . . . . . . .
8 | 3 1 1 . . . . . . . . . . .
10 | 1 1 . . 1 . . . . . . . . .
12 | 2 1 1 1 . 1 . . . . . . . .
14 | 1 1 . . . . . . . . . . . .
16 | 4 1 1 . . . 1 . . . . . . .
18 | 1 1 . 1 . . . 1 . . . . . .
20 | 2 1 2 . 1 . . . . . . . . .
22 | 1 1 . . . . . . 1 . . . . .
24 | 3 1 1 1 . 1 . . . . . . . .
26 | 1 1 . . . . . . . . . . . .
28 | 2 1 1 . . . . . . 1 . . . .
30 | 1 1 . 1 1 . . . . . 1 . . .
32 | 5 1 1 . . . 1 . . . . . . .
34 | 1 1 . . . . . . . . . . . .
36 | 2 1 1 1 . 1 . 1 . . . 1 . .
38 | 1 1 . . . . . . . . . . . .
40 | 3 1 2 . 1 . . . . . . . 1 .
42 | 1 1 . 2 . . . . . . . . . 1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..1201
EXAMPLE
In A214406, row number 4 is:
(k=0) (k=1) (k=2) (k=3) (k=4)
1 112 718 744 105
Now,
(2*4-0)* 1 + (0+1)*112 = 120
(2*4-1)*112 + (1+1)*718 = 2220
(2*4-2)*718 + (2+1)*744 = 6540
(2*4-3)*744 + (3+1)*105 = 4140
(2*4-4)*105 + (4+1)* 0 = 420
The GCD of {120, 2220, 6540, 4140, 420} is 60, so a(4)=60.
MATHEMATICA
T[n_, k_]:=T[n, k]=If[n==0&&k==0, 1, If[n==0||k<0||k>n, 0, (4*n-2*k-1)*T[n-1, k-1]+(2*k+1)*T[n-1, k]]]
A[n_]:=Table[(2*n-k)*T[n, k]+(k+1)*T[n, k+1], {k, 0, n}]/.{List->GCD}
Table[A[n], {n, 1, 100}]
PROG
(PARI)
r(n)=if(n==0, 1, (r(n-1)*x/(1-x^2))')
a(n)=my(p=(r(n))'*(1-x^2)^(2*n+1)/2); p/factorback(factor(p))
for(n=1, 60, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Luc Rousseau, Nov 23 2020
STATUS
approved