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Triangle T read by rows: coefficients of polynomials generating array A099597.
2

%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