cp's OEIS Frontend

A263634 Irregular triangle read by rows: row n gives coefficients of n-th logarithmic polynomial L_n(x_1, x_2, ...) with monomials sorted into standard order.

%I A263634 #165 Apr 04 2025 21:25:36
%S A263634 1,-1,1,2,-3,1,-6,12,-4,-3,1,24,-60,20,30,-5,-10,1,-120,360,-120,-270,
%T A263634 30,120,30,-6,-15,-10,1,720,-2520,840,2520,-210,-1260,-630,42,210,140,
%U A263634 210,-7,-21,-35,1
%N A263634 Irregular triangle read by rows: row n gives coefficients of n-th logarithmic polynomial L_n(x_1, x_2, ...) with monomials sorted into standard order.
%C A263634 "Standard order" here means as produced by Maple's "sort" command.
%C A263634 From _Petros Hadjicostas_, May 27 2020: (Start)
%C A263634 According to the Maple help files for the "sort" command, polynomials in multiple variables are "sorted in total degree with ties broken by lexicographic order (this is called graded lexicographic order)."
%C A263634 Thus for example, x_1^2*x_3 = x_1*x_1*x_3 > x_1*x_2*x_2 = x_1*x_2^2, while x_1^2*x_4 = x_1*x_1*x_4 > x_1*x_2*x_3. (End)
%C A263634 Row sums are 0 (for n > 1). Numbers of terms in rows are partition numbers A000041.
%C A263634 From _Tom Copeland_, Nov 06 2015: (Start)
%C A263634 With the formal Taylor series f(x) = 1 + x[1] x + x[2] x^2/2! + ... , the partition polynomials of this entry give d[log(f(x))]/dx = L_1(x[1]) + L_2(x[1], x[2]) x + L_3(...) x^2/2! + ..., and the coefficients of the reduced polynomials with x[n] = t are signed A028246.
%C A263634 The raising operator R = x + d[log(f(D)]/dD = x + L_1(x[1]) + L_2[x[1], x[2]) D + L_3(x[1], x[2], x[3]) D^2/2! + ... with D = d/dx generates an Appell sequence of polynomials, given umbrally by P_n(x[1], ..., x[n]; x) = (x[.] + x)^n = Sum_{k=0..n} binomial(n,k) x[k] * x^(n-k) = R^n 1 with the e.g.f. f(t)*e^(x*t) = exp[t P.(x[1], ..., x[.]; x)]. P_0 = x[0] = 1.
%C A263634 The umbral compositional inverse Appell sequence is generated by R = x - d[log(f(D))]/dD with e.g.f. e^(x*t)/f(t) = exp[t IP.(x[1], ..., x[.]; x)], so umbrally IP_n(x[1], ..., x[n]; P.(x[1], ..., x[n]; x)) = x^n = P_n(x[1], ..., x[n]; IP.(x[1], ..., x[n]; x)). An unsigned array for the reduced IP_n(x[1], ..., x[n]; x) polynomials with IP_0 = x[0] = 1 and x[n] = -1 for n > 0 is A154921, for which f(t) = 2 - e^t. (End)
%C A263634 From _Tom Copeland_, Sep 08 2016: (Start)
%C A263634 The Appell formalism allows a matrix representation in the power basis x^n of the raising operator R that incorporates this array's partition polynomials L_n(x[1], ..., x[n]):
%C A263634 VP_(n+1) = VP_n * R = VP_n * XPS^(-1) * MX * XPS, where XPS is the matrix formed from multiplying the n-th diagonal of the Pascal matrix PS of A007318 by the indeterminate x[n], with x[0] = 1 for the main diagonal of ones, i.e., XPS[n,k] = PS[n,k] * x[n-k]; the matrix MX is A129185; the matrix XPS^(-1) is the inverse of XPS, which can be formed by multiplying the diagonals of the Pascal matrix by the partition polynomials IPT(n, x[1], ..., x[n]) of A133314, i.e., XPS^(-1)[n,k] = PS[n,k] * IPT(n-k, x[1], ...); and VP_n is the row vector in the power basis representing the Appell polynomial P_n(x) formed from the basic sequence of moments 1, x[1], x[2], ..., i.e., umbrally P_n(x) = (x[.] + x)^n = Sum_{k=0..n} binomial(n,k) * x[k] * x^(n-k).
%C A263634 Then R = XPS^(-1) * MX * XPS is the Pascal matrix PS with an additional first superdiagonal of ones and the other lower diagonals multiplied by the partition polynomials of this array, i.e., R[n,k] = PS[n,k] * L_{n+1-k}(x[1], ..., x[n+1-k]) except for the first superdiagonal of ones.
%C A263634 Consistently, VP_n = (1, 0, 0, ...) * R^n = (1, 0, 0, ...) * XPS^(-1) * MX^n * XPS = (1, 0, 0, ...) * MX^n * XPS = the n-th row vector of XPS, which is the vector representation of P_n(x) = (x[.] + x)^n with x[0] = 1.
%C A263634 See the Copeland link for the umbral representation R = exp[g.*D] * x * exp[h.*D] that reflects the matrix representations.
%C A263634 The Stirling partition polynomials of the first kind St1_n(a[1], a[2], ..., a[n]) of A036039, the Stirling partition polynomials of the second kind St2_n(b[1], b[2], ..., b[n]) of A036040, and the refined Lah polynomials Lah_n[c[1], c[2], ..., c[n]) of A130561 are Appell sequences in the respective distinguished indeterminates a[1], b[1], and c[1]. Comparing the formulas for their raising operators with that in this entry, L_n(x[1], x[2], ..., x[n]) evaluates to
%C A263634 A) (n-1)! * a[n] for x[n] = St1_n(a[1], a[2], ..., a[n]);
%C A263634 B) b[n] for x[n] = St2_n(b[1], b[2], ..., b[n]);
%C A263634 C) n! * c[n] for x[n] = Lah_n(c[1], c[2], ..., c[n]).
%C A263634 Conversely, from the respective e.g.f.s (added Sep 12 2016)
%C A263634 D) x[n] = St1_n(L_1(x[1])/0!, ..., L_n(x[1], ..., x[n])/(n-1)!);
%C A263634 E) x[n] = St2_n(L_1(x[1]), ..., L_n(x[1], ..., x[n]));
%C A263634 F) x[n] = Lah_n(L_1(x[1])/1!, ..., L_n(x[1], ..., x[n])/n!).
%C A263634 Given only the Appell sequence with no closed form for the e.g.f., the raising operator can be generated using this formalism, as has been partially done for A134264. (End)
%C A263634 For the Appell sequences above, the raising operator is related to the recursion P_(n+1)(x) = x * P_n(x) + Sum_{k=0..n} binomial(n,k) * L_(n-k+1)(x[1], ..., x[n+k-1]) * P_k(x). For a derivation and connections to formal cumulants (c_n = L_n(x[1], ...)) and moments (m_n = x[n]), see the Copeland link on noncrossing partitions. With x = 0, the recursion reduces to x[n+1] =  Sum_{k = 0..n} binomial(n,k) * L_(n-k+1)(x[1], ..., x[n+k-1]) * x[k] with x[0] = 1. This array is a differently ordered version of A127671. - _Tom Copeland_, Sep 13 2016
%C A263634 With x[n] = x^(n-1), a signed version of A130850 is obtained. - _Tom Copeland_, Nov 14 2016
%C A263634 See p. 2 of Getzler for a relation to stable graphs called necklaces used in computations for Deligne-Mumford-Knudsen moduli spaces of stable curves of genus 1. - _Tom Copeland_, Nov 15 2019
%C A263634 For a relation to a combinatorial Faa di Bruno Hopf algebra related to functional composition, as presented by Connes and Moscovici, see Figueroa et al. - _Tom Copeland_, Jan 17 2020
%C A263634 From _Tom Copeland_, May 17 2020: (Start)
%C A263634 The e.g.f. of an Appell sequence is f(t) e^(x*t) with f(0) = 1. Given the Laguerre-Polya class function f(t) = e^(-a*t^2 + b*t) Product_m (1 - t/z_m) e^(t/z_m) with a = 0 for simplicity (more generally a >= 0) and b real and where the product runs over all the zeros z_m of f(t) with all zeros real and Sum_m 1/(z_m)^2 convergent, the raising operator of the Appell polynomials is R = x + b - Sum_{k > 0} c_(k+1) D^k with c_k = Sum_m (1/(z_m)^k), i.e., traces of powers of the reciprocals of the zeros. From R in earlier comments, b = L_1(x_1) and otherwise c_k = -L_k(x_1, ..., x_k).
%C A263634 The Laguerre / Turan / de Gua inequalities (Csordas and Williamson and Skovgaard) imply that all the zeros of each Appell polynomial are real and simple and its extrema are local maxima above the x-axis and local minima below and are located above or below the zeros of the next lower degree Appell polynomial. (End)
%C A263634 From _Tom Copeland_, Oct 15 2020: (Start)
%C A263634 With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
%C A263634 A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
%C A263634 B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
%C A263634 C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
%C A263634 Expansions of log(f(x)) are given in
%C A263634 I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
%C A263634 II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
%C A263634 Expansions of exp(f(x)-1) are given in
%C A263634 III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
%C A263634 IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
%C A263634 V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
%C A263634 Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)
%C A263634 Ignoring signs, these polynomials appear in Schröder in the set of equations (II) on p. 343 and in Stewart's translation on p. 31. - _Tom Copeland_, Aug 25 2021
%D A263634 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 140, 156, 308.
%H A263634 Peter Luschny, <a href="/A263634/b263634.txt">Row n for n = 1..20</a>.
%H A263634 Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/12/23/appell-ops-cumulants-noncrossing-partitions-dyck-lattice-paths-and-inversion/">Appell polynomials, cumulants, noncrossing partitions, Dyck paths, and inversion</a>, 2014.
%H A263634 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/11/21/the-creation-raising-operators-for-appell-sequences/">The creation / raising operators for Appell sequences</a>, 2015.
%H A263634 G. Csordas and J. Williamson, <a href="https://doi.org/10.1090/S0002-9939-1975-0361017-4">The zeros of the Jensen polynomials are simple</a>, Proceed. of the AMS, 49(1) (1975), 263-264.
%H A263634 H. Figueroa, J. Gracia-Bondia, and J. Varilly, <a href="https://arxiv.org/abs/math/0508337">Faa di Bruno Hopf algebras</a>, arXiv:0508337 [math.CO], 2005; see p. 3.
%H A263634 E. Getzler, <a href="http://arxiv.org/abs/alg-geom/9612005">The semi-classical approximation for modular operads</a>, arXiv:alg-geom/9612005, 1996; see p. 2.
%H A263634 J. Novak and M. LaCroix, <a href="https://arxiv.org/abs/1205.2097">Three lectures on free probability</a>, arXiv:1205.2097 [math.CO], 2012.
%H A263634 E. Schröder, <a href="https://www.semanticscholar.org/paper/Ueber-unendlich-viele-Algorithmen-zur-Aufl%C3%B6sung-der-Schr%C3%B6der/7d37b4dbf960770e926575456b9504f5e785b048">Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen</a>, Mathematische Annalen vol. 2, 317-365, 1870.
%H A263634 H. Skovgaard, <a href="https://www.mscand.dk/article/view/10396">On inequalities of the Turan type</a>, Math. Scand. 2 (1954), 65-73.
%H A263634 G. Stewart, <a href="https://drum.lib.umd.edu/handle/1903/577">On infinitely many algorithms for solving equations</a>, 1993, (translation into English of Schröder's paper above)
%F A263634 G.f.: Log(1 + Sum_{i >= 1} x_i*t^i/i!) =  Sum_{n >= 1} L_n(x_1, x_2, ...)*t^n/n!. [Comtet, p. 140, Eq. [5a]. - corrected by _Tom Copeland_, Sep 08 2016]
%F A263634 Conjecture: row polynomials are R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) - Sum_{j=1..n-1} binomial(n-2,j-1)*R(j,k)*R(n-j,1) for n > 1, k > 0 with R(1,k) = x_k for k > 0. - _Mikhail Kurkov_, Mar 30 2025
%e A263634 The first few polynomials are:
%e A263634 (1) x[1].
%e A263634 (2) -x[1]^2 + x[2].
%e A263634 (3) 2*x[1]^3 - 3*x[1]*x[2] + x[3].
%e A263634 (4) -6*x[1]^4 + 12*x[1]^2*x[2] - 4*x[1]*x[3] - 3*x[2]^2 + x[4].
%e A263634 (5) 24*x[1]^5 - 60*x[1]^3*x[2] + 20*x[1]^2*x[3] + 30*x[1]*x[2]^2 - 5*x[1]*x[4] - 10*x[2]*x[3] + x[5].
%e A263634 (6) -120*x[1]^6 + 360*x[1]^4*x[2] - 120*x[1]^3*x[3] - 270*x[1]^2*x[2]^2 + 30*x[1]^2*x[4] + 120*x[1]*x[2]*x[3] + 30*x[2]^3 - 6*x[1]*x[5] - 15*x[2]*x[4] - 10*x[3]^2 + x[6].
%e A263634 ...
%e A263634 [1]    1
%e A263634 [2]   -1,    1
%e A263634 [3]    2,   -3,     1
%e A263634 [4]   -6,   12,    -4,    -3,   1
%e A263634 [5]   24,  -60,    20,    30,  -5,  -10,   1
%e A263634 [6] -120,  360,  -120,  -270,  30,  120,  30, -6, -15, -10, 1
%p A263634 triangle := proc(numrows) local E, s, Q;
%p A263634 E := add(x[i]*t^i/i!, i=1..numrows);
%p A263634 s := series(log(1 + E), t, numrows+1);
%p A263634 Q := k -> sort(expand(k!*coeff(s, t, k)));
%p A263634 seq(print(coeffs(Q(k))), k=1..numrows) end:
%p A263634 triangle(6); # updated by _Peter Luschny_, May 27 2020
%Y A263634 Cf. A000041, A178867, A263633, A028246, A007318, A129185, A133314, A154921.
%Y A263634 Cf. A036039, A036040, A127671, A130561, A130850, A134264.
%Y A263634 Cf. A263916.
%K A263634 sign,tabf
%O A263634 1,4
%A A263634 _N. J. A. Sloane_, Oct 29 2015