A292605 - OEIS

%I #19 Mar 24 2020 13:24:22

%S 1,1,0,19,1,0,1513,166,1,0,315523,52715,1361,1,0,136085041,30543236,

%T 1528806,10916,1,0,105261234643,29664031413,2257312622,42421946,87375,

%U 1,0,132705221399353,45011574747714,4637635381695,153778143100,1156669095,699042,1,0

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

%C See the comments in A292604.

%F 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 Triangle starts:

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

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

%e [0][         1]

%e [1][         1,        0]

%e [2][        19,        1,        0]

%e [3][      1513,      166,        1,     0]

%e [4][    315523,    52715,     1361,     1,  0]

%e [5][ 136085041, 30543236,  1528806, 10916,  1, 0]

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

%p A292605_row := proc(n) if n = 0 then return [1] fi;

%p add(A278073(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:

%p for n from 0 to 6 do A292605_row(n) od;

%o (Sage) # uses[A278073_row from A278073]

%o def A292605_row(n):

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

%o     L = A278073_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(A292605_row(n))

%Y F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} is this triangle, F_{4} = A292606.

%Y First column: A002115. Row sums: A014606. Alternating row sums: A292609.

%Y Cf. A181985, A278073.

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Sep 20 2017