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%I #15 Nov 13 2019 06:46:06
%S 1,1,1,1,0,2,1,9,-12,6,1,-104,204,-120,24,1,2265,-4840,3540,-1080,120,
%T 1,-71064,164910,-138840,54360,-10080,720,1,3079825,-7626948,7134330,
%U -3300360,808920,-100800,5040,1,-176449776,460982648,-468313104,244938960,-72266880,12156480,-1088640,40320
%N Triangle T read by rows: coefficients of polynomials generating array A099597.
%C Row sums are k (A000027), left edge columns are factorials (A000142). [_Peter Bala_, Aug 19 2013]
%F The row polynomials satisfy the second order recurrence equation R(n,x) = (n*x+1)*R(n-1,x-1) - (n-1)*(x-1)*R(n-2,x-2), with the initial conditions R(0,x) = 1 and R(1,x) = 1+x. - _Peter Bala_, Aug 19 2013
%e 1
%e x + 1
%e 2*x^2 + 1
%e 6*x^3 - 12*x^2 + 9*x + 1
%e 24*x^4 - 120*x^3 + 204*x^2 - 104*x + 1
%e 120*x^5 - 1080*x^4 + 3540*x^3 - 4840*x^2 + 2265*x + 1
%p # Define row polynomials R(n, x) recursively:
%p R := proc(n, x) option remember; if n = 0 then 1 elif n = 1 then 1+x
%p else (n*x+1)*procname(n-1,x-1) - (n-1)*(x-1)*procname(n-2, x-2) fi end:
%p Trow := n -> PolynomialTools:-CoefficientList(R(n,x), x);
%p seq(Trow(n), n = 0..10); # _Peter Bala_, Aug 19 2013
%t R[n_, x_] := R[n, x] = (n x + 1) R[n-1, x-1] - (n-1) (x-1) R[n-2, x-2]; R[0, _] = 1; R[1, x_] = 1 + x;
%t Table[CoefficientList[R[n, x], x], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Nov 13 2019 *)
%K sign,tabl
%O 0,6
%A _Ralf Stephan_, Oct 28 2004
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