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A335256 Irregular triangle read by rows: row n gives the coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ..., x_n) with monomials sorted into standard order.

%I A335256 #53 Mar 21 2025 04:32:06
%S A335256 1,1,1,1,3,1,1,6,4,3,1,1,10,10,15,5,10,1,1,15,20,45,15,60,15,6,15,10,
%T A335256 1,1,21,35,105,35,210,105,21,105,70,105,7,21,35,1,1,28,56,210,70,560,
%U A335256 420,56,420,280,840,105,28,168,280,210,280,8,28,56,35,1
%N A335256 Irregular triangle read by rows: row n gives the coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ..., x_n) with monomials sorted into standard order.
%C A335256 "Standard order" means as produced by Maple's "sort" command.
%C A335256 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 A335256 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.
%C A335256 The number of terms in the n-th row is A000041(n), while the sum of the terms is A000110(n).
%C A335256 The function Bell(n,k) in the PARI program below is a modification of a similar function in the PARI help files and uses the Faà di Bruno formula (cf. A036040).
%D A335256 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 134 and 307-310.
%D A335256 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, Chapter 2, Section 8 and table on page 49.
%H A335256 E. T. Bell, <a href="https://www.jstor.org/stable/1967979">Partition polynomials</a>, Ann. Math., 29 (1927-1928), 38-46.
%H A335256 E. T. Bell, <a href="https://www.jstor.org/stable/1968431">Exponential polynomials</a>, Ann. Math., 35 (1934), 258-277.
%H A335256 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F A335256 B_n(x[1], ..., x[n]) = Sum_{k=1..n} B_{n,k}(x[1], ..., x[n-k+1]), where B_{n,k} = B_{n,k}(x[1], ..., x[n-k+1]) are the partial exponential Bell polynomials that satisfy B_{n,1} = x[n] for n >= 1 and B_{n,k} = (1/k)*Sum_{m=k-1..n-1} binomial(n,m)*x[n-m]*B_{m,k-1} for n >= 2 and k = 2..n.
%F A335256 E.g.f.: Exp(Sum_{i >= 1} x_i*t^i/i!) = 1 + Sum_{n >= 1} B_n(x_1, x_2, ..., x_n)*t^n/n! [Comtet, p. 134, Eq. [3b]].
%e A335256 The first few complete exponential Bell polynomials are:
%e A335256 (1) x[1];
%e A335256 (2) x[1]^2 + x[2];
%e A335256 (3) x[1]^3 + 3*x[1]*x[2] + x[3];
%e A335256 (4) x[1]^4 + 6*x[1]^2*x[2] + 4*x[1]*x[3] + 3*x[2]^2 + x[4];
%e A335256 (5) x[1]^5 + 10*x[1]^3*x[2] + 10*x[1]^2*x[3] + 15*x[1]*x[2]^2 + 5*x[1]*x[4] + 10*x[2]*x[3] + x[5];
%e A335256 (6) x[1]^6 + 15*x[1]^4*x[2] + 20*x[1]^3*x[3] + 45*x[1]^2*x[2]^2 + 15*x[1]^2*x[4] + 60*x[1]*x[2]*x[3] + 15*x[2]^3 + 6*x[1]*x[5] + 15*x[2]*x[4] + 10*x[3]^2 + x[6].
%e A335256 (7) x[1]^7 + 21*x[1]^5*x[2] + 35*x[1]^4*x[3] + 105*x[1]^3*x[2]^2 + 35*x[1]^3*x[4] + 210*x[1]^2*x[2]*x[3] + 105*x[1]*x[2]^3 + 21*x[1]^2*x[5] + 105*x[1]*x[2]*x[4] + 70*x[1]*x[3]^2 + 105*x[2]^2*x[3] + 7*x[1]*x[6] + 21*x[2]*x[5] + 35*x[3]*x[4] + x[7].
%e A335256 ...
%e A335256 The first few rows of the triangle are
%e A335256   1;
%e A335256   1,  1;
%e A335256   1,  3,  1;
%e A335256   1,  6,  4,   3,   1;
%e A335256   1, 10, 10,  15,   5,  10,   1;
%e A335256   1, 15, 20,  45,  15,  60,  15,  6,  15, 10,   1;
%e A335256   1, 21, 35, 105,  35, 210, 105, 21, 105, 70, 105, 7, 21, 35, 1;
%e A335256   ...
%p A335256 triangle := proc(numrows) local E, s, Q;
%p A335256 E := add(x[i]*t^i/i!, i=1..numrows);
%p A335256 s := series(exp(E), t, numrows+1);
%p A335256 Q := k -> sort(expand(k!*coeff(s, t, k)));
%p A335256 seq(print(coeffs(Q(k))), k=1..numrows) end:
%p A335256 triangle(8); # _Peter Luschny_, May 30 2020
%t A335256 imax = 10;
%t A335256 polys = (CoefficientList[Exp[Sum[x[i]*t^i/i!, {i, 1, imax}]] + O[t]^imax // Normal, t]*Range[0, imax-1]!) // Rest;
%t A335256 Table[MonomialList[polys[[i]], Array[x, i], "DegreeLexicographic"] /. x[_] -> 1, {i, 1, imax-1}] // Flatten (* _Jean-François Alcover_, Jun 02 2024 *)
%o A335256 (PARI) /* It produces the partial exponential Bell polynomials in decreasing degree, but the monomials are not necessarily in standard order. */
%o A335256 Bell(n,k)= { my(x, v, dv, var = i->eval(Str("X", i))); v = vector(n, i, if (i==1, 'E, var(i-1))); dv = vector(n, i, if (i==1, 'X*var(1)*'E, var(i))); x = diffop('E, v, dv, n) / 'E; if (k < 0, subst(x,'X, 1), polcoeff(x, k, 'X)); };
%o A335256 row(n) = for(k=1, n, print1("[", Bell(n, n+1-k), "]", ","))
%Y A335256 For different versions, see A178867 and A268441.
%Y A335256 Cf. A000041, A000110, A036040.
%K A335256 nonn,tabf
%O A335256 1,5
%A A335256 _Petros Hadjicostas_, May 28 2020