A339100 a(n) = GCD({(2*n-k)*T(n,k)+(k+1)*T(n,k+1), k=0..n}), where T(n,k) stands for A214406 (the second-order Eulerian numbers of type B).

1, 6, 1, 60, 1, 42, 1, 120, 1, 66, 1, 5460, 1, 6, 1, 4080, 1, 798, 1, 3300, 1, 138, 1, 10920, 1, 6, 1, 1740, 1, 14322, 1, 8160, 1, 6, 1, 3838380, 1, 6, 1, 270600, 1, 12642, 1, 1380, 1, 282, 1, 371280, 1, 66, 1, 3180, 1, 798, 1, 3480, 1, 354, 1, 567867300

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

Cf. A214406, A165886.

Sequence in context: A083837 A049213 A165886 * A344918 A174502 A056218

Adjacent sequences: A339097 A339098 A339099 * A339101 A339102 A339103

Keyword

nonn

Author

Luc Rousseau, Nov 23 2020

Status

approved