A246656 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.
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, 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, -513, 0, 182, 0, -22, 0, 1, 0, 1, 1865, 344, -2686, -484, 945, 175, -114, -21, 5, 1, 0
Offset: 0
Examples
The first few polynomials and their coefficients:
0; 0;
1, 0; 1;
0, 1, 0; x;
1, 1, 1, 0; x*(x+1)+1;
0, 1, 0, 1, 0; x*(x^2+1);
1, 1, 2, 2, 1, 0; x*(x+1)*(x^2+x+1)+1;
0, 3, 0, -1, 0, 1, 0; x*(x^4-x^2+3);
1, 8, 5, -5, 0, 3, 1, 0; x*(x+1)*(x^4+2*x^3-2*x^2-3*x+8)+1;
0,-18, 0, 29, 0, -8, 0, 1,0; x*(x^6-8*x^4+29*x^2-18);
The values of some polynomials:
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n: -4 -3 -2 -1 0 1 2 3
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p_0(n): 0, 0, 0, 0, 0, 0, 0, 0, A000004
p_1(n): 1, 1, 1, 1, 1, 1, 1, 1, A000012
p_2(n): -4, -3, -2, -1, 0, 1, 2, 3, A001477
p_3(n): 13, 7, 3, 1, 1, 3, 7, 13, A002061
p_4(n): -68, -30, -10, -2, 0, 2, 10, 30, A034262
p_5(n): 157, 43, 7, 1, 1, 7, 43, 157,
p_6(n): -972, -225, -30, -3, 0, 3, 30, 225,
Programs
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Maple
with(Student[NumericalAnalysis]): poly := proc(n) local B; if n = 0 then return 0 fi; B := (n,k) -> round(evalf(2*(BesselK(n,2)*BesselI(k,2) -(-1)^(n+k)*BesselI(n,2)*BesselK(k,2)),64)); [seq([k+iquo(n,2),B(k+n,k)], k=-iquo(n,2)..n-1)]; PolynomialInterpolation(%, independentvar=x); expand(Interpolant(%)) end: A246656_row := n -> seq(coeff(poly(n),x,j), j=0..n); seq(print(A246656_row(n)), n=0..11);