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A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

%I A292605 #19 Mar 24 2020 13:24:22
%S A292605 1,1,0,19,1,0,1513,166,1,0,315523,52715,1361,1,0,136085041,30543236,
%T A292605 1528806,10916,1,0,105261234643,29664031413,2257312622,42421946,87375,
%U A292605 1,0,132705221399353,45011574747714,4637635381695,153778143100,1156669095,699042,1,0
%N A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).
%C A292605 See the comments in A292604.
%F A292605 F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) for n>0 and F_{3; 0}(x) = 1.
%e A292605 Triangle starts:
%e A292605 [n\k][       0         1         2       3  4  5]
%e A292605 --------------------------------------------------
%e A292605 [0][         1]
%e A292605 [1][         1,        0]
%e A292605 [2][        19,        1,        0]
%e A292605 [3][      1513,      166,        1,     0]
%e A292605 [4][    315523,    52715,     1361,     1,  0]
%e A292605 [5][ 136085041, 30543236,  1528806, 10916,  1, 0]
%p A292605 Coeffs := f -> PolynomialTools:-CoefficientList(expand(f),x):
%p A292605 A292605_row := proc(n) if n = 0 then return [1] fi;
%p A292605 add(A278073(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
%p A292605 for n from 0 to 6 do A292605_row(n) od;
%o A292605 (Sage) # uses[A278073_row from A278073]
%o A292605 def A292605_row(n):
%o A292605     if n == 0: return [1]
%o A292605     L = A278073_row(n)
%o A292605     S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
%o A292605     return expand(S).list() + [0]
%o A292605 for n in (0..5): print(A292605_row(n))
%Y A292605 F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} is this triangle, F_{4} = A292606.
%Y A292605 First column: A002115. Row sums: A014606. Alternating row sums: A292609.
%Y A292605 Cf. A181985, A278073.
%K A292605 nonn,tabl
%O A292605 0,4
%A A292605 _Peter Luschny_, Sep 20 2017