| Displaying 1-3 of 3 results found.
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1, 1, 18, 1348, 264168, 107059696, 77812181280, 92178659860288, 166177428088123008, 432941319641569590016, 1565619431839802755158528, 7608371278068863387781342208, 48386147164823804551330131929088
MAPLE
a := proc(n) A292605_row(n); add((-1)^k*%[k+1], k=0..n) end: seq(a(n), n=0..12);
Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).
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4
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
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}
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.
REFERENCES
G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.
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.
EXAMPLE
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]
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]];
PROG
(Sage)
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))
Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).
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3
1, 1, 0, 69, 1, 0, 33661, 988, 1, 0, 60376809, 2669683, 16507, 1, 0, 288294050521, 17033188586, 212734266, 261626, 1, 0, 3019098162602349, 223257353561605, 4297382231090, 17634518610, 4196345, 1, 0
FORMULA
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) for n>0 and F_{4; 0}(x) = 1.
EXAMPLE
Triangle starts:
[n\k][ 0 1 2 3 4 5]
--------------------------------------------------
[0] [ 1]
[1] [ 1, 0]
[2] [ 69, 1, 0]
[3] [ 33661, 988, 1, 0]
[4] [ 60376809, 2669683, 16507, 1, 0]
[5] [288294050521, 17033188586, 212734266, 261626, 1, 0]
MAPLE
Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292606_row := proc(n) if n = 0 then return [1] fi;
add( A278074(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end: for n from 0 to 6 do A292606_row(n) od;
PROG
if n == 0: return [1]
S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
return expand(S).list() + [0]
for n in (0..5): print( A292606_row(n))
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