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 A225117 Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order. 9
 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_{k=0..n}(-1)^k*T(n,k), 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. 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. 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) 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 ... 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 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. = 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)] (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 (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 CROSSREFS Coefficients of A_{n,1}(x) = A008292, coefficients of A_{n,2}(x) = A060187, coefficients of A_{n,4}(x) = A225118. Cf. A173018, A123125, A284861. Sequence in context: A358501 A323493 A052661 * A088624 A309878 A066409 Adjacent sequences: A225114 A225115 A225116 * A225118 A225119 A225120 KEYWORD nonn,tabl AUTHOR Peter Luschny, May 02 2013 STATUS approved

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