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A225117 Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order. 4
1, 2, 1, 4, 13, 1, 8, 93, 60, 1, 16, 545, 1131, 251, 1, 32, 2933, 14498, 10678, 1018, 1, 64, 15177, 154113, 262438, 88998, 4089, 1, 128, 77101, 1475736, 4890287, 3870352, 692499, 16376, 1, 256, 388321, 13270807, 77404933, 117758659, 50476003, 5175013, 65527, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The row sums equal the triple factorial numbers A032031 and the alternating row sums, i.e., sum((-1)^k*T(n,k),k=0..n), are up to a sign A000810. - Johannes W. Meijer, May 04 2013

LINKS

Table of n, a(n) for n=0..44.

Peter Luschny, Generalized Eulerian polynomials.

FORMULA

Generating function of the polynomials is gf(n, k) = k^n*n!*(1/x-1)^(n+1)[t^n](x*e^(t*x/k)*(1-x*e(t*x))^(-1)) for k = 3; here [t^n]f(t,x) is the coefficient of t^n in f(t,x).

EXAMPLE

[0]  1

[1]  2*x   +   1

[2]  4*x^2 +  13*x   +    1

[3]  8*x^3 +  93*x^2 +   60*x   +   1

[4] 16*x^4 + 545*x^3 + 1131*x^2 + 251*x + 1

MAPLE

gf := proc(n, k) local f; f := (x, t) -> x*exp(t*x/k)/(1-x*exp(t*x));

series(f(x, t), t, n+2); ((1-x)/x)^(n+1)*k^n*n!*coeff(%, t, n):

collect(simplify(%), x) end:

seq(print(seq(coeff(gf(n, 3), x, n-k), k=0..n)), n=0..6);

# Recurrence

P := proc(n, x) option remember; if n = 0 then 1 else

  (n*x+(1/3)*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x);

  expand(%) fi end:

A225117 := (n, k) -> 3^n*coeff(P(n, x), x, n-k):

seq(print(seq(A225117(n, k), k=0..n)), n=0..5);  # Peter Luschny, Mar 08 2014

MATHEMATICA

gf[n_, k_] := Module[{f, s}, f[x_, t_] := x*Exp[t*x/k]/(1-x*Exp[t*x]); s = Series[f[x, t], {t, 0, n+2}]; ((1-x)/x)^(n+1)*k^n*n!*SeriesCoefficient[s, {t, 0, n}]]; Table[Table[SeriesCoefficient[gf[n, 3], {x, 0, n-k}], {k, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Maple *)

PROG

(Sage)

@CachedFunction

def EB(n, k, x):  # Modified cardinal B-splines

    if n == 1: return 0 if (x < 0) or (x >= 1) else 1

    return k*x*EB(n-1, k, x) + k*(n-x)*EB(n-1, k, x-1)

def EulerianPolynomial(n, k): # Generalized Eulerian polynomials

    R.<x> = ZZ[]

    if x == 0: return 1

    return add(EB(n+1, k, m+1/k)*x^m for m in (0..n))

[EulerianPolynomial(n, 3).coefficients()[::-1] for n in (0..5)]

CROSSREFS

Coefficients of A_{n,1}(x) = A008292, coefficients of A_{n,2}(x) = A060187, coefficients of A_{n,4}(x) = A225118.

Sequence in context: A205870 A064297 A052661 * A088624 A066409 A006173

Adjacent sequences:  A225114 A225115 A225116 * A225118 A225119 A225120

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 02 2013

STATUS

approved

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Last modified March 30 20:18 EDT 2017. Contains 284302 sequences.