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A292606 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).

%I A292606 #25 Mar 24 2020 12:36:56
%S A292606 1,1,0,69,1,0,33661,988,1,0,60376809,2669683,16507,1,0,288294050521,
%T A292606 17033188586,212734266,261626,1,0,3019098162602349,223257353561605,
%U A292606 4297382231090,17634518610,4196345,1,0
%N A292606 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).
%C A292606 See the comments in A292604.
%F A292606 F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) for n>0 and F_{4; 0}(x) = 1.
%e A292606 Triangle starts:
%e A292606 [n\k][          0            1          2       3   4   5]
%e A292606 --------------------------------------------------
%e A292606 [0] [           1]
%e A292606 [1] [           1,           0]
%e A292606 [2] [          69,           1,         0]
%e A292606 [3] [       33661,         988,         1,      0]
%e A292606 [4] [    60376809,     2669683,     16507,      1,  0]
%e A292606 [5] [288294050521, 17033188586, 212734266, 261626,  1,  0]
%p A292606 Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
%p A292606 A292606_row := proc(n) if n = 0 then return [1] fi;
%p A292606 add(A278074(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
%p A292606 for n from 0 to 6 do A292606_row(n) od;
%o A292606 (Sage) # uses[A278074_row from A278074]
%o A292606 def A292606_row(n):
%o A292606     if n == 0: return [1]
%o A292606     L = A278074_row(n)
%o A292606     S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
%o A292606     return expand(S).list() + [0]
%o A292606 for n in (0..5): print(A292606_row(n))
%Y A292606 F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} = A292605, F_{4} is this triangle.
%Y A292606 First column: A211212. Row sums: A014608. Alternating row sums: A292607.
%Y A292606 Cf. A181985.
%K A292606 nonn,tabl
%O A292606 0,4
%A A292606 _Peter Luschny_, Sep 26 2017