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%I #29 Mar 06 2020 04:05:17
%S 1,1,0,5,1,0,61,28,1,0,1385,1011,123,1,0,50521,50666,11706,506,1,0,
%T 2702765,3448901,1212146,118546,2041,1,0,199360981,308869464,
%U 147485535,24226000,1130235,8184,1,0
%N Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).
%C The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:
%C F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.
%C F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.
%C F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.
%C F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.
%C F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.
%C The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.
%C Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:
%C F_{m} : F_{0} F_{1} F_{2} F_{3} F_{4}
%C x = -1: A165326 A155585 A002105 A292609 A292607
%C x = 1: A000012 A000142 A000680 A014606 A014608 ... (m*n)!/m!^n
%C x = 0: -- A000012 A000364 A002115 A211212 ... m-alternating permutations of length m*n.
%C Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.
%D G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.
%F F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.
%e Triangle starts:
%e [n\k][ 0 1 2 3 4 5 6]
%e --------------------------------------------------
%e [0][ 1]
%e [1][ 1, 0]
%e [2][ 5, 1, 0]
%e [3][ 61, 28, 1, 0]
%e [4][ 1385, 1011, 123, 1, 0]
%e [5][ 50521, 50666, 11706, 506, 1, 0]
%e [6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]
%p Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
%p A292604_row := proc(n) if n = 0 then return [1] fi;
%p add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
%p for n from 0 to 6 do A292604_row(n) od;
%t T[n_, k_] /; 1 <= k <= n := T[n, k] = k (2 k - 1) T[n - 1, k - 1] + k^2 T[n - 1, k]; T[_, 1] = 1; T[_, _] = 0;
%t F[2, 0][_] = 1; F[2, n_][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
%t row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
%t Table[row[n], {n, 0, 7}] (* _Jean-François Alcover_, Jul 06 2019 *)
%o (Sage)
%o def A292604_row(n):
%o if n == 0: return [1]
%o S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
%o return expand(S).list() + [0]
%o for n in (0..6): print(A292604_row(n))
%Y F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.
%Y First column: A000364. Row sums: A000680. Alternating row sums: A002105.
%Y Cf. A181985, A241171.
%K nonn,tabl
%O 0,4
%A _Peter Luschny_, Sep 20 2017
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