A292604 - OEIS

%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