A292606 - OEIS

%I #25 Mar 24 2020 12:36:56

%S 1,1,0,69,1,0,33661,988,1,0,60376809,2669683,16507,1,0,288294050521,

%T 17033188586,212734266,261626,1,0,3019098162602349,223257353561605,

%U 4297382231090,17634518610,4196345,1,0

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

%C See the comments in A292604.

%F 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 Triangle starts:

%e [n\k][          0            1          2       3   4   5]

%e --------------------------------------------------

%e [0] [           1]

%e [1] [           1,           0]

%e [2] [          69,           1,         0]

%e [3] [       33661,         988,         1,      0]

%e [4] [    60376809,     2669683,     16507,      1,  0]

%e [5] [288294050521, 17033188586, 212734266, 261626,  1,  0]

%p Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):

%p A292606_row := proc(n) if n = 0 then return [1] fi;

%p add(A278074(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:

%p for n from 0 to 6 do A292606_row(n) od;

%o (Sage) # uses[A278074_row from A278074]

%o def A292606_row(n):

%o     if n == 0: return [1]

%o     L = A278074_row(n)

%o     S = sum(L[k]*(x-1)^(n-k) for k in (0..n))

%o     return expand(S).list() + [0]

%o for n in (0..5): print(A292606_row(n))

%Y F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} = A292605, F_{4} is this triangle.

%Y First column: A211212. Row sums: A014608. Alternating row sums: A292607.

%Y Cf. A181985.

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Sep 26 2017