A076256 - OEIS

%I #32 Sep 23 2024 18:05:31

%S 1,0,-2,-2,0,6,0,24,0,-24,24,0,-240,0,120,0,-720,0,2400,0,-720,-720,0,

%T 15120,0,-25200,0,5040,0,40320,0,-282240,0,282240,0,-40320,40320,0,

%U -1451520,0,5080320,0,-3386880,0,362880,0,-3628800,0,43545600,0,-91445760,0,43545600,0,-3628800

%N Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the constant term.

%C Let T(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2).

%C The denominators are (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.

%H Andrew Howroyd, <a href="/A076256/b076256.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%F T(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k) for n + k even;

%F T(n, k) = 0 for n + k odd.

%F 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 From _Fabian Pereyra_, Sep 11 2024: (Start)

%F 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 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 Then, the p(n,x) satisfy:

%F p(n,x) = -2*n*x*p(n-1,x) - n*(n-1)*(1+x^2)*p(n-2,x).

%F p'(n,x) = -n*(n+1)*p(n-1,x).

%F (1+x^2)*p''(n,x) - 2*n*x*p'(n,x) + n*(n+1)*p(n,x) = 0.

%F 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 Sum_{k=0..n} abs(T(n,k)) = A000165(n). - _Alois P. Heinz_, Sep 18 2024

%e Triangle begins:

%e      1;

%e      0,    -2;

%e     -2,     0,     6;

%e      0,    24,     0,     -24;

%e     24,     0,  -240,       0,    120;

%e      0,  -720,     0,    2400,      0,   -720;

%e   -720,     0, 15120,       0, -25200,      0, 5040;

%e      0, 40320,     0, -282240,      0, 282240,    0, -40320;

%e    ...

%t 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 (PARI) T(n,k) = if((n+k)%2, 0, (-1)^((n+k)/2)*n!*binomial(n+1, k)) \\ _Andrew Howroyd_, Aug 08 2024

%Y Cf. A000165, A007318, A076257, A076741, A076743.

%K sign,tabl,easy

%O 0,3

%A _Mohammad K. Azarian_, Nov 05 2002

%E Edited by _Dean Hickerson_, Nov 28 2002