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A292604
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Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).
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4
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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
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OFFSET
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0,4
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COMMENTS
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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.
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REFERENCES
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G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.
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LINKS
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FORMULA
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F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.
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EXAMPLE
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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]
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MAPLE
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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;
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MATHEMATICA
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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]];
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PROG
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(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))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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