A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

1, 1, 0, 5, 1, 0, 61, 28, 1, 0, 1385, 1011, 123, 1, 0, 50521, 50666, 11706, 506, 1, 0, 2702765, 3448901, 1212146, 118546, 2041, 1, 0, 199360981, 308869464, 147485535, 24226000, 1130235, 8184, 1, 0

Offset: 0

Author

Peter Luschny, Sep 20 2017

Keywords

• nonn
• tabl

Comments

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:

F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.

F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.

F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.

F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.

The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.

Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:

F_{m} : F_{0} F_{1} F_{2} F_{3} F_{4}

x = -1: A165326 A155585 A002105 A292609 A292607

x = 1: A000012 A000142 A000680 A014606 A014608 ... (m*n)!/m!^n

x = 0: -- A000012 A000364 A002115 A211212 ... m-alternating permutations of length m*n.

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.

Examples

			Triangle starts:
[n\k][    0        1        2       3     4  5  6]
--------------------------------------------------
[0][      1]
[1][      1,       0]
[2][      5,       1,       0]
[3][     61,      28,       1,      0]
[4][   1385,    1011,     123,      1,    0]
[5][  50521,   50666,   11706,    506,    1, 0]
[6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]
		

References

• G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.

Crossrefs

F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.

First column: A000364. Row sums: A000680. Alternating row sums: A002105.

Cf. A181985, A241171.

Programs

Maple

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292604_row := proc(n) if n = 0 then return [1] fi;
add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292604_row(n) od;

Mathematica

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;
F[2, 0][] = 1; F[2, n][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
Table[row[n], {n, 0, 7}] (* Jean-François Alcover, Jul 06 2019 *)

Sage

def A292604_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..6): print(A292604_row(n))

Formula

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.