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%I #194 Apr 19 2023 09:55:34
%S 1,0,1,0,1,1,0,1,4,1,0,1,11,11,1,0,1,26,66,26,1,0,1,57,302,302,57,1,0,
%T 1,120,1191,2416,1191,120,1,0,1,247,4293,15619,15619,4293,247,1,0,1,
%U 502,14608,88234,156190,88234,14608,502,1,0,1,1013,47840,455192,1310354
%N Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
%C The beginning of this sequence does not quite agree with the usual version, which is A173018. - _N. J. A. Sloane_, Nov 21 2010
%C Each row of A123125 is the reverse of the corresponding row in A173018. - _Michael Somos_, Mar 17 2011
%C A008292 (subtriangle for k>=1 and n>=1 is the main entry for these numbers.
%C Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,...] where DELTA is the operator defined in A084938.
%C Row sums are the factorials. - _Roger L. Bagula_ and _Gary W. Adamson_, Aug 14 2008
%C If the initial zero column is deleted, the result is A008292. - _Roger L. Bagula_ and _Gary W. Adamson_, Aug 14 2008
%C This result gives an alternative method of calculating the Eulerian numbers by an Umbral Calculus expansion from Comtet. - _Roger L. Bagula_, Nov 21 2009
%C This function seems to be equivalent to the PolyLog expansion. - _Roger L. Bagula_, Nov 21 2009
%C A raising operator formed from the e.g.f. of this entry is the generator of a sequence of polynomials p(n,x;t) defined in A046802 that specialize to those for A119879 as p(n,x;-1), A007318 as p(n,x;0), A073107 as p(n,x;1), and A046802 as p(n,0;t). See Copeland link for more associations. - _Tom Copeland_, Oct 20 2015
%C The Eulerian numbers in this setup count the permutation trees of power n and width k (see the Luschny link). For the associated combinatorial statistic over permutations see the Sage program below and the example section. - _Peter Luschny_, Dec 09 2015 [See Elder et al. link. _Peter Luschny_, Jul 13 2022]
%C From _Wolfdieter Lang_, Apr 03 2017: (Start)
%C The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k are the numerator polynomials of the o.g.f. G(n, x) of n-powers {m^n}_{m>=0} (with 0^0 = 1): G(n, x) = R(n, x)/(1-x)^(n+1). See the Aug 14 2008 formula, where f(x,n) = R(n, x). The e.g.f. of R(n, t) is given in Copeland's Oct 14 2015 formula below.
%C The first nine column sequences are A000007, A000012, A000295, A000460, A000498, A000505, A000514, A001243, A001244. (End)
%C With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of this entry, A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - _Tom Copeland_, Jan 24 2020
%C Let b(n) = (1/(n+1))*Sum_{k=0..n-1} (-1)^(n-k+1)*T(n, k+1) / binomial(n, k+1). Then b(n) = Bernoulli(n, 1) = -n*Zeta(1 - n) = Integral_{x=0..1} F_n(x) for n >= 1. Here F_n(x) are the signed Fubini polynomials (A278075). (See also Rzadkowski and Urlinska, example 1.) - _Peter Luschny_, Feb 15 2021
%C Patrick J. Burchell (see link) describes the following method: To get the k-th row of the triangle write the nonnegative integers with a fixed exponent k as a sequence, 0^k, 1^k, 2^k, ..., and then apply the first differences to them k + 1 times. - _Peter Luschny_, Apr 02 2023
%D L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245. [_Roger L. Bagula_, Nov 21 2009]
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed.; Addison-Wesley, 1994, p. 268, Row reversed table 268. - _Wolfdieter Lang_, Apr 03 2017
%D Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91. - _Roger L. Bagula_ and _Gary W. Adamson_, Aug 14 2008
%H Reinhard Zumkeller, <a href="/A123125/b123125.txt">Rows n = 0..125 of triangle, flattened</a>
%H Paul Barry, <a href="http://arxiv.org/abs/1105.3043">Eulerian polynomials as moments, via exponential Riordan arrays</a>, arXiv preprint arXiv:1105.3043 [math.CO], 2011, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry7/barry172.html">J. Int. Seq. 14 (2011) # 11.9.5</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry1/barry263.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations</a>, Journal of Integer Sequences, 17 (2014), #14.2.3.
%H Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H Paul Barry, <a href="https://arxiv.org/abs/1803.10297">Generalized Eulerian Triangles and Some Special Production Matrices</a>, arXiv:1803.10297 [math.CO], 2018.
%H V. Batyrev and M. Blume, <a href="http://arxiv.org/abs/0911.3607">The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces</a>, p. 11, arXiv:/0911.3607 [math.AG], 2009. [_Tom Copeland_, Oct 16 2015]
%H Anna Borowiec and Wojciech Mlotkowski, <a href="http://arxiv.org/abs/1509.03758">New Eulerian numbers of type D</a>, arXiv:1509.03758 [math.CO], 2015.
%H Patrick J. Burchell, <a href="https://doi.org/10.48550/arXiv.2303.14045">A Generalisation of Ramanujan's (back of the envelope) Method for Divergent Series</a>, arXiv:2303.14045 math.NT, 2023.
%H A. Cohen, <a href="https://pure.tue.nl/ws/portalfiles/portal/2928709/Metis225320.pdf">Eulerian polynomials of spherical type</a>, Münster J. of Math. 1 (2008). [_Tom Copeland_, Oct 16 2015]
%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera</a>
%H Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker and Amanda Welch, <a href="https://doi.org/10.48550/arxiv.2206.13409">Homomesies on permutations -- an analysis of maps and statistics in the FindStat database</a>, math.CO, arXiv, 2022. (Def. 4.20 and Prop. 4.22.)
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000021">The number of descents of a permutation.</a>
%H F. Hirzebruch, <a href="https://www.uni-muenster.de/FB10/mjm/vol_1/mjm_vol_1_02.pdf">Eulerian polynomials</a>, Münster J. of Math. 1 (2008), pp. 9-12.
%H P. Hitczenko and S. Janson, <a href="http://arxiv.org/abs/1212.5498">Weighted random staircase tableaux</a>, arXiv preprint arXiv:1212.5498 [math.CO], 2012.
%H Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, <a href="https://arxiv.org/abs/1807.01412">An asymptotic distribution theory for Eulerian recurrences with applications</a>, arXiv:1807.01412 [math.CO], 2018.
%H Svante Janson, <a href="http://arxiv.org/abs/1305.3512">Euler-Frobenius numbers and rounding</a>, arXiv preprint arXiv:1305.3512 [math.PR], 2013.
%H Katarzyna Kril and Wojciech Mlotkowski, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p27">Permutations of Type B with Fixed Number of Descents and Minus Signs</a>, Volume 26(1) of The Electronic Journal of Combinatorics, 2019.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%H Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 22.
%H A. Losev and Y. Manin, <a href="http://arxiv.org/abs/math/0001003">New moduli spaces of pointed curves and pencils of flat connections</a>, arXiv preprint arXiv:math/0001003 [math.AG], 2000 (p. 8). - _Tom Copeland_, Oct 16 2015
%H Peter Luschny, <a href="http://www.oeis.org/wiki/User:Peter_Luschny/PermutationTrees">Permutation Trees</a>
%H G. Rzadkowski and M. Urlinska, <a href="http://arxiv.org/abs/1612.06635">A Generalization of the Eulerian Numbers</a>, arXiv:1612.06635 [math.CO], 2016.
%F Sum_{k=0..n} T(n,k) = n! = A000142(n).
%F Sum_{k=0..n} 2^k*T(n,k) = A000629(n).
%F Sum_{k=0..n} 3^k*T(n,k) = abs(A009362(n+1)).
%F Sum_{k=0..n} 2^(n-k)*T(n,k) = A000670(n).
%F Sum_{k=0..n} T(n,k)*3^(n-k) = A122704(n). - _Philippe Deléham_, Nov 07 2007
%F G.f.: f(x,n) = (1 - x)^(n + 1)*Sum_{k>=0} k^n*x^k. - _Roger L. Bagula_ and _Gary W. Adamson_, Aug 14 2008. f is not the g.f. of the triangle, it is the polynomial of row n. See an Apr 03 2017 comment above - _Wolfdieter Lang_, Apr 03 2017
%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000629(n), A123227(n), A201355(n), A201368(n) for x = 0, 1, 2, 3, 4, 5 respectively. - _Philippe Deléham_, Dec 01 2011
%F E.g.f. (1-t)/(1-t*exp((1-t)x)). A123125 * A007318 = A130850 = unsigned A075263, related to reversed A028246. A007318 * A123125 = A046802. Evaluating the row polynomials at -1, giving the alternating-sign row sum, generates A009006. - _Tom Copeland_, Oct 14 2015
%F From _Wolfdieter Lang_, Apr 03 2017: (Start)
%F T(n, k) = A173018(n, n-k), 0 <= k <= n. Row reversed Euler's triangle. See Graham et al., p. 268.
%F Recurrence (from A173018): T(n, 0) = 1 if n=0 else 0; T(n, k) = 0 if n < k and T(n, k) = (n+1-k)*T(n-1, k-1) + k*T(n-1, k) else.
%F T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n-j, k-j)*S2(n, j)*j!, 0 <= k <= n, else 0. For S2(n, k)*k! see A131689.
%F The recurrence for the o.g.f. of the sequence of column k is
%F G(k, x) = (x/(1 - k*x))*(E_x - (k-2))*G(k-1, x), with the Euler operator E_x = x*d_x, for k >= 1, with G(0, x) = 1. (Proof from the recurrence of T(n, k)).
%F The e.g.f of the sequence of column k is found from E(k, x) = (1 + int(A(k, x),x)*exp(-k*x))*exp(k*x), k >= 1, with the recurrence
%F A(k, x) = x*A(k-1, x) +(1 + (1-k)*(1-x))*E(k-1, x) for k >= 1, with A(0,x)= 0. (Proof from the recurrence of T(n, k)). (End)
%F T(n, k) = Sum_{j=0..n-k} (-1)^j*(n-j-k+1)^n*binomial(n + 1, j). - _Peter Luschny_, Aug 12 2022
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10...
%e 0: 1
%e 1: 0 1
%e 2: 0 1 1
%e 3: 0 1 4 1
%e 4: 0 1 11 11 1
%e 5: 0 1 26 66 26 1
%e 6: 0 1 57 302 302 57 1
%e 7: 0 1 120 1191 2416 1191 120 1
%e 8: 0 1 247 4293 15619 15619 4293 247 1
%e 9: 0 1 502 14608 88234 156190 88234 14608 502 1
%e 10: 0 1 1013 47840 455192 1310354 1310354 455192 47840 1013 1
%e ... Reformatted. - _Wolfdieter Lang_, Feb 14 2015
%e ------------------------------------------------------------------
%e The width statistic over permutations, n=4.
%e [1, 2, 3, 4] => 3; [1, 2, 4, 3] => 2; [1, 3, 2, 4] => 2; [1, 3, 4, 2] => 2;
%e [1, 4, 2, 3] => 2; [1, 4, 3, 2] => 1; [2, 1, 3, 4] => 3; [2, 1, 4, 3] => 2;
%e [2, 3, 1, 4] => 2; [2, 3, 4, 1] => 3; [2, 4, 1, 3] => 2; [2, 4, 3, 1] => 2;
%e [3, 1, 2, 4] => 3; [3, 1, 4, 2] => 3; [3, 2, 1, 4] => 2; [3, 2, 4, 1] => 3;
%e [3, 4, 1, 2] => 3; [3, 4, 2, 1] => 2; [4, 1, 2, 3] => 4; [4, 1, 3, 2] => 3;
%e [4, 2, 1, 3] => 3; [4, 2, 3, 1] => 3; [4, 3, 1, 2] => 3; [4, 3, 2, 1] => 2;
%e Gives row(4) = [0, 1, 11, 11, 1]. - _Peter Luschny_, Dec 09 2015
%e ------------------------------------------------------------------
%e From _Wolfdieter Lang_, Apr 03 2017: (Start)
%e Recurrence: T(5, 3) = (6-3)*T(4, 2) + 3*T(4, 3) = 3*11 + 3*11 = 66.
%e O.g.f. column k=2: (x/(1 - 2*x))*E_x*(x/(1-x) = (x/1-x)^2/(1-2*x).
%e E.g.f. column k=2: A(2, x) = x*A(1, x) + x*E(1, x) = x*1 + x*(exp(x)-1) = x*exp(x), hence E(2, x) = (1 + int(x*exp(-x),x ))*exp(2*x) = exp(x)*(exp(x) - (1+x)). See A000295. (End)
%p gf := 1/(1 - t*exp(x)): ser := series(gf, x, 12):
%p cx := n -> (-1)^(n + 1)*factor(n!*coeff(ser, x, n)*(t - 1)^(n + 1)):
%p seq(print(seq(coeff(cx(n), t, k), k = 0..n)), n = 0..9); # _Peter Luschny_, Feb 11 2021
%p A123125 := proc(n, k) option remember; if k = n then 1 elif k <= 0 or k > n then 0 else k*procname(n-1, k) + (n-k+1)*procname(n-1, k-1) fi end:
%p seq(print(seq(A123125(n, k), k=0..n)), n=0..10); # _Peter Luschny_, Mar 28 2021
%p # Alternative (Patrick J. Burchell):
%p t := a -> Statistics:-Difference([0, a]): Trow := k -> (t@@(k+1))([seq(n^k, n = 0..k)]):
%p seq(print(Trow(n)), n = 0..6); # _Peter Luschny_, Apr 02 2023
%t f[x_, n_] := f[x, n] = (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}];
%t Table[CoefficientList[f[x, n], x], {n,0,9}] // Flatten (* Roger L. Bagula, Aug 14 2008 *)
%t t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = (n-k) t[n-1, k-1] + (k+1) t[n-1, k]; T[n_, k_] := t[n, n-k];
%t Table[T[n, k], {n,0,10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 26 2019 *)
%t A123125[n_, k_] := Sum[(-1)^j*(n - j - k + 1)^n * Binomial[n + 1, j], {j, 0, n - k}];
%t Table[A123125[n, k], {n, 0, 9}, {k, 0, n}] // TableForm (* _Peter Luschny_, Aug 12 2022 *)
%o (Haskell)
%o a123125 n k = a123125_tabl !! n !! k
%o a123125_row n = a123125_tabl !! n
%o a123125_tabl = [1] : zipWith (:) [0, 0 ..] a008292_tabl
%o -- _Reinhard Zumkeller_, Nov 06 2013
%o (Sage)
%o def statistic_eulerian(pi):
%o if not pi: return 0
%o h, i, branch, next = 0, len(pi), [0], pi[0]
%o while True:
%o while next < branch[len(branch)-1]:
%o del(branch[len(branch)-1])
%o current = 0
%o h += 1
%o while next > current:
%o i -= 1
%o if i == 0: return h
%o branch.append(next)
%o current, next = next, pi[i]
%o def A123125_row(n):
%o L = [0]*(n+1)
%o for p in Permutations(n):
%o L[statistic_eulerian(p)] += 1
%o return L
%o [A123125_row(n) for n in range(7)] # _Peter Luschny_, Dec 09 2015
%Y See A008292 (subtriangle for k>=1 and n>=1), which is the main entry for these numbers. Another version has the zeros at the ends of the rows, as in Concrete Mathematics: see A173018.
%Y Cf. A007318, A130850, A028246, A046802, A009006, A019538, A090582, A119879, A248727.
%K nonn,easy,tabl
%O 0,9
%A _Philippe Deléham_, Sep 30 2006
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