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%I #13 Mar 19 2023 08:28:22
%S 1,2,0,3,2,0,4,8,8,0,5,20,56,56,0,6,40,216,608,608,0,7,70,616,3352,
%T 9440,9440,0,8,112,1456,12928,70400,198272,198272,0,9,168,3024,39696,
%U 352768,1921152,5410688,5410688,0,10,240,5712,103872,1364800,12129664,66057856,186043904,186043904,0
%N Triangle of coefficients of a certain sequence of polynomials f_n(x) arising in connection with deformations of coordinate rings of type D Kleinian singularities.
%C f_n(x) has the property that whenever (a,b) is a pair of complex numbers satisfying 2ab = a^2 + 2a + b^2 + 2b, we have f_n(a) + f_n(b) = 2(a^n - b^n)/(a-b) (interpreted as 2na^(n-1) if a=b). Using the pairs (0,0), (0,-2), (-2,-6), (-6,-12), (-12,-20), ...(see A002378), this enables us to successively deduce the values of f_n(0), f_n(-2),... (and this of course determines f_n(x)). There may be no simpler characterization.
%H Paul Boddington, <a href="http://wrap.warwick.ac.uk/3482/">No-cycle algebras and representation theory</a>, Ph.D. thesis, University of Warwick, 2004.
%e The array begins
%e 1
%e 2 0
%e 3 2 0
%e 4 8 8 0
%e corresponding to the polynomials f_1(x) = 1, f_2(x) = 2x, f_3(x) = 3x^2 + 2x, f_4(x) = 4x^3 + 8x^2 + 8x.
%o (PARI) f(n, a, b) = if (a==b, 2*n*a^(n-1), 2*(a^n - b^n)/(a-b));
%o row(n) = if (n==1, return([1])); my(v = vector(n-1, k, f(n, -(k-1)*k, -k*(k+1)))); my(m = matrix(n-1, n-1, i, j, (-j*(j-1))^i + (-(j+1)*j)^i)); concat(Vecrev(v/m), 0); \\ _Michel Marcus_, Mar 19 2023
%Y Cf. A002378, A005439 (right diagonal).
%K nonn,tabl
%O 1,2
%A _Paul Boddington_, Aug 20 2004
%E Typo corrected and extended by _Michel Marcus_, Mar 19 2023
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