|
%I #12 Sep 16 2014 05:47:32
%S 0,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,2,2,1,0,0,3,0,-1,0,1,0,1,8,5,-5,0,
%T 3,1,0,0,-18,0,29,0,-8,0,1,0,1,-80,-13,121,29,-35,-7,4,1,0,0,357,0,
%U -513,0,182,0,-22,0,1,0,1,1865,344,-2686,-484,945,175,-114,-21,5,1,0
%N Triangle read by rows: T(n, k) is the coefficient of x^k of the polynomial p_n(x) representing the n-th diagonal of A246654.
%e The first few polynomials and their coefficients:
%e 0; 0;
%e 1, 0; 1;
%e 0, 1, 0; x;
%e 1, 1, 1, 0; x*(x+1)+1;
%e 0, 1, 0, 1, 0; x*(x^2+1);
%e 1, 1, 2, 2, 1, 0; x*(x+1)*(x^2+x+1)+1;
%e 0, 3, 0, -1, 0, 1, 0; x*(x^4-x^2+3);
%e 1, 8, 5, -5, 0, 3, 1, 0; x*(x+1)*(x^4+2*x^3-2*x^2-3*x+8)+1;
%e 0,-18, 0, 29, 0, -8, 0, 1,0; x*(x^6-8*x^4+29*x^2-18);
%e The values of some polynomials:
%e ------------------------------------------------
%e n: -4 -3 -2 -1 0 1 2 3
%e ------------------------------------------------
%e p_0(n): 0, 0, 0, 0, 0, 0, 0, 0, A000004
%e p_1(n): 1, 1, 1, 1, 1, 1, 1, 1, A000012
%e p_2(n): -4, -3, -2, -1, 0, 1, 2, 3, A001477
%e p_3(n): 13, 7, 3, 1, 1, 3, 7, 13, A002061
%e p_4(n): -68, -30, -10, -2, 0, 2, 10, 30, A034262
%e p_5(n): 157, 43, 7, 1, 1, 7, 43, 157,
%e p_6(n): -972, -225, -30, -3, 0, 3, 30, 225,
%p with(Student[NumericalAnalysis]):
%p poly := proc(n) local B; if n = 0 then return 0 fi;
%p B := (n,k) -> round(evalf(2*(BesselK(n,2)*BesselI(k,2)
%p -(-1)^(n+k)*BesselI(n,2)*BesselK(k,2)),64));
%p [seq([k+iquo(n,2),B(k+n,k)], k=-iquo(n,2)..n-1)];
%p PolynomialInterpolation(%, independentvar=x);
%p expand(Interpolant(%)) end:
%p A246656_row := n -> seq(coeff(poly(n),x,j), j=0..n);
%p seq(print(A246656_row(n)), n=0..11);
%Y Cf. A246654, A001477, A002061, A034262.
%K tabl,sign
%O 0,18
%A _Peter Luschny_, Sep 13 2014
|
