OFFSET
0,3
COMMENTS
Let T(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2).
The denominators are (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FORMULA
T(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k) for n + k even;
T(n, k) = 0 for n + k odd.
E.g.f.: A(x,t) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*t^k/n! = 1/(1 + 2*x*t + x^2*(1+t^2)). - Fabian Pereyra, Aug 08 2024
From Fabian Pereyra, Sep 11 2024: (Start)
T(n,k) = -n*(n-1)*T(n-2,k) - 2*n*T(n-1,k-1) - n*(n-1)*T(n-2,k-2), with T(0,0) = 1, T(n,k) = 0 if k<0 or k>n.
Let p(n,x) the n-th polynomial in x defined by: p(n,x) = Sum_{k=0..n} T(n,k)*x^k.
Then, the p(n,x) satisfy:
p(n,x) = -2*n*x*p(n-1,x) - n*(n-1)*(1+x^2)*p(n-2,x).
p'(n,x) = -n*(n+1)*p(n-1,x).
(1+x^2)*p''(n,x) - 2*n*x*p'(n,x) + n*(n+1)*p(n,x) = 0.
Integral_{x=-inf..inf} p(n,x)*p(m,x)*(1/(1+x^2))^(max(n,m)+1) dx = n!*(n+1)!*pi* delta(n,m), where delta(n,m) is the Kronecker delta. (End)
Sum_{k=0..n} abs(T(n,k)) = A000165(n). - Alois P. Heinz, Sep 18 2024
EXAMPLE
Triangle begins:
1;
0, -2;
-2, 0, 6;
0, 24, 0, -24;
24, 0, -240, 0, 120;
0, -720, 0, 2400, 0, -720;
-720, 0, 15120, 0, -25200, 0, 5040;
0, 40320, 0, -282240, 0, 282240, 0, -40320;
...
MATHEMATICA
a[n_, k_] := Coefficient[Expand[Together[(1+x^2)^(n+1)*D[1/(1+x^2), {x, n}]]], x, k]; Flatten[Table[a[n, k], {n, 0, 10}, {k, 0, n}]]
PROG
(PARI) T(n, k) = if((n+k)%2, 0, (-1)^((n+k)/2)*n!*binomial(n+1, k)) \\ Andrew Howroyd, Aug 08 2024
CROSSREFS
KEYWORD
AUTHOR
Mohammad K. Azarian, Nov 05 2002
EXTENSIONS
Edited by Dean Hickerson, Nov 28 2002
STATUS
approved