cp's OEIS Frontend

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).

%I A339100 #16 Dec 14 2021 00:32:12
%S A339100 1,6,1,60,1,42,1,120,1,66,1,5460,1,6,1,4080,1,798,1,3300,1,138,1,
%T A339100 10920,1,6,1,1740,1,14322,1,8160,1,6,1,3838380,1,6,1,270600,1,12642,1,
%U A339100 1380,1,282,1,371280,1,66,1,3180,1,798,1,3480,1,354,1,567867300
%N 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).
%C A339100 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.
%C A339100 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:
%C A339100   \ p|
%C A339100    \ |  2  3  5  7 11 13 17 19 23 29 31 37 41 43
%C A339100   n \|
%C A339100   ---+-------------------------------------------
%C A339100    2 |  1  1  .  .  .  .  .  .  .  .  .  .  .  .
%C A339100    4 |  2  1  1  .  .  .  .  .  .  .  .  .  .  .
%C A339100    6 |  1  1  .  1  .  .  .  .  .  .  .  .  .  .
%C A339100    8 |  3  1  1  .  .  .  .  .  .  .  .  .  .  .
%C A339100   10 |  1  1  .  .  1  .  .  .  .  .  .  .  .  .
%C A339100   12 |  2  1  1  1  .  1  .  .  .  .  .  .  .  .
%C A339100   14 |  1  1  .  .  .  .  .  .  .  .  .  .  .  .
%C A339100   16 |  4  1  1  .  .  .  1  .  .  .  .  .  .  .
%C A339100   18 |  1  1  .  1  .  .  .  1  .  .  .  .  .  .
%C A339100   20 |  2  1  2  .  1  .  .  .  .  .  .  .  .  .
%C A339100   22 |  1  1  .  .  .  .  .  .  1  .  .  .  .  .
%C A339100   24 |  3  1  1  1  .  1  .  .  .  .  .  .  .  .
%C A339100   26 |  1  1  .  .  .  .  .  .  .  .  .  .  .  .
%C A339100   28 |  2  1  1  .  .  .  .  .  .  1  .  .  .  .
%C A339100   30 |  1  1  .  1  1  .  .  .  .  .  1  .  .  .
%C A339100   32 |  5  1  1  .  .  .  1  .  .  .  .  .  .  .
%C A339100   34 |  1  1  .  .  .  .  .  .  .  .  .  .  .  .
%C A339100   36 |  2  1  1  1  .  1  .  1  .  .  .  1  .  .
%C A339100   38 |  1  1  .  .  .  .  .  .  .  .  .  .  .  .
%C A339100   40 |  3  1  2  .  1  .  .  .  .  .  .  .  1  .
%C A339100   42 |  1  1  .  2  .  .  .  .  .  .  .  .  .  1
%H A339100 Antti Karttunen, <a href="/A339100/b339100.txt">Table of n, a(n) for n = 1..1201</a>
%e A339100 In A214406, row number 4 is:
%e A339100 (k=0) (k=1) (k=2) (k=3) (k=4)
%e A339100   1    112   718   744   105
%e A339100 Now,
%e A339100 (2*4-0)*  1 + (0+1)*112 =  120
%e A339100 (2*4-1)*112 + (1+1)*718 = 2220
%e A339100 (2*4-2)*718 + (2+1)*744 = 6540
%e A339100 (2*4-3)*744 + (3+1)*105 = 4140
%e A339100 (2*4-4)*105 + (4+1)*  0 =  420
%e A339100 The GCD of {120, 2220, 6540, 4140, 420} is 60, so a(4)=60.
%t A339100 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]]]
%t A339100 A[n_]:=Table[(2*n-k)*T[n,k]+(k+1)*T[n,k+1],{k,0,n}]/.{List->GCD}
%t A339100 Table[A[n],{n,1,100}]
%o A339100 (PARI)
%o A339100 r(n)=if(n==0,1,(r(n-1)*x/(1-x^2))')
%o A339100 a(n)=my(p=(r(n))'*(1-x^2)^(2*n+1)/2);p/factorback(factor(p))
%o A339100 for(n=1,60,print1(a(n),", "))
%Y A339100 Cf. A214406, A165886.
%K A339100 nonn
%O A339100 1,2
%A A339100 _Luc Rousseau_, Nov 23 2020