A225117 Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.
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
Offset: 0
Examples
[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 ... The triangle T(n, k) begins: n \ k 0 1 2 3 4 5 6 7 ... 0: 1 1: 2 1 2: 4 13 1 3: 8 93 60 1 5: 16 545 1131 251 1 6: 32 2933 14498 10678 1018 1 7: 64 15177 154113 262438 88998 4089 1 8: 128 77101 1475736 4890287 3870352 692499 16376 1 ... - _Wolfdieter Lang_, Apr 08 2017 Three term recurrence: T(2,1) = (3*(2-1)+1)*2 + (3*1+2)*1 = 13. - _Wolfdieter Lang_, Apr 10 2017
Links
- Peter Luschny, Generalized Eulerian polynomials.
- Zhe Wang and Zhi-Yong Zhu, The spiral property of q-Eulerian numbers of type B, The Australasian Journal of Combinatorics, Volume 87(1) (2023), Pages 198-202. See p. 199.
Crossrefs
Programs
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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
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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 *) -
PARI
T(n, k) = sum(j=0, n - k, (-1)^(n - k - j)*binomial(n + 1, n - k - j)*(1 + 3*j)^n); for(n=0, 10, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 10 2017
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Python
from sympy import binomial def T(n,k): return sum((-1)**(n - k - j)* binomial(n + 1, n - k - j)*(1 + 3*j)**n for j in range(n - k + 1)) for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 10 2017
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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.= 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)]
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).
From Wolfdieter Lang, Apr 10 2017: (Start)
T(n, k) = Sum_{j=0..n-k} (-1)^(n-k-j)*binomial(n+1, n-k-j)*(1+3*j)^n, 0 <= k <= n.
T(n, k) = Sum_{m=0..n-k} (-1)^(n-k-m)*binomial(n-m, k)*A284861(n, m), 0 <= k <= n.
The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k are R(n, x) = (x-1)^n*Sum_{m=0} A284861(n, m)*(1/(x-1))^m, n >= 0, i.e. the row polynomials of A284861 in the variable 1/(x-1) multiplied by (x-1)^n.
The row polynomials with falling powers are P(n, x) = (1-x)^n*Sum_{m=0..n} A284861(n, m)*(x/(1-x))^m, n >= 0.
The e.g.f. of the row polynomials in falling powers of x (A_{n, 3}(x) of the name) is exp((1-x)*z)/(1 - (x/(1 - x)) * (exp(3*(1-x)*z) - 1)) = (1-x)*exp((1-x)*z)/(1 - x*exp(3*(1-x)*z)).
The e.g.f. of the row polynomials R(n, x) (rising powers of x) is then (1-x)*exp(2*(1-x)*z)/(1 - x*exp(3*(1-x)*z)).
Three term recurrence: T(n, k) = 0 if n < k , T(n, -1) = 0, T(0,0) = 1, T(n, k) = (3*(n-k)+1)*T(n-1, k-1) + (3*k+2)*T(n-1, k) for n >= 1, k=0..n. (End)
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