This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A076256 #32 Sep 23 2024 18:05:31
%S A076256 1,0,-2,-2,0,6,0,24,0,-24,24,0,-240,0,120,0,-720,0,2400,0,-720,-720,0,
%T A076256 15120,0,-25200,0,5040,0,40320,0,-282240,0,282240,0,-40320,40320,0,
%U A076256 -1451520,0,5080320,0,-3386880,0,362880,0,-3628800,0,43545600,0,-91445760,0,43545600,0,-3628800
%N A076256 Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the constant term.
%C A076256 Let T(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2).
%C A076256 The denominators are (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.
%H A076256 Andrew Howroyd, <a href="/A076256/b076256.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A076256 T(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k) for n + k even;
%F A076256 T(n, k) = 0 for n + k odd.
%F A076256 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
%F A076256 From _Fabian Pereyra_, Sep 11 2024: (Start)
%F A076256 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.
%F A076256 Let p(n,x) the n-th polynomial in x defined by: p(n,x) = Sum_{k=0..n} T(n,k)*x^k.
%F A076256 Then, the p(n,x) satisfy:
%F A076256 p(n,x) = -2*n*x*p(n-1,x) - n*(n-1)*(1+x^2)*p(n-2,x).
%F A076256 p'(n,x) = -n*(n+1)*p(n-1,x).
%F A076256 (1+x^2)*p''(n,x) - 2*n*x*p'(n,x) + n*(n+1)*p(n,x) = 0.
%F A076256 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)
%F A076256 Sum_{k=0..n} abs(T(n,k)) = A000165(n). - _Alois P. Heinz_, Sep 18 2024
%e A076256 Triangle begins:
%e A076256 1;
%e A076256 0, -2;
%e A076256 -2, 0, 6;
%e A076256 0, 24, 0, -24;
%e A076256 24, 0, -240, 0, 120;
%e A076256 0, -720, 0, 2400, 0, -720;
%e A076256 -720, 0, 15120, 0, -25200, 0, 5040;
%e A076256 0, 40320, 0, -282240, 0, 282240, 0, -40320;
%e A076256 ...
%t A076256 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}]]
%o A076256 (PARI) T(n,k) = if((n+k)%2, 0, (-1)^((n+k)/2)*n!*binomial(n+1, k)) \\ _Andrew Howroyd_, Aug 08 2024
%Y A076256 Cf. A000165, A007318, A076257, A076741, A076743.
%K A076256 sign,tabl,easy
%O A076256 0,3
%A A076256 _Mohammad K. Azarian_, Nov 05 2002
%E A076256 Edited by _Dean Hickerson_, Nov 28 2002