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A268441 Triangle read by rows, the coefficients of the Bell polynomials.

%I A268441 #33 May 27 2020 20:17:19
%S A268441 1,0,1,0,1,1,0,1,3,1,0,1,3,4,6,1,0,1,10,5,15,10,10,1,0,1,10,15,6,15,
%T A268441 60,15,45,20,15,1,0,1,35,21,7,105,70,105,21,105,210,35,105,35,21,1,0,
%U A268441 1,35,56,28,8,280,210,280,168,28,105,840,280,420,56,420,560,70,210,56,28,1
%N A268441 Triangle read by rows, the coefficients of the Bell polynomials.
%C A268441 The triangle of coefficients of the inverse Bell polynomials is A268442.
%D A268441 L. Comtet, Advanced combinatorics, The art of finite and infinite expansions, 1974.
%H A268441 Peter Luschny, <a href="/A268441/b268441.txt">First 26 rows, flattened</a>
%H A268441 E. T. Bell, <a href="https://www.jstor.org/stable/1967979">Partition polynomials</a>, Ann. Math., 29 (1927-1928), 38-46.
%H A268441 E. T. Bell, <a href="https://www.jstor.org/stable/1968431">Exponential polynomials</a>, Ann. Math., 35 (1934), 258-277.
%H A268441 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%F A268441 E.g.f.: exp( Sum_{k>=1} x_{k}*t^k/k! ), monomials in negative lexicographic order.
%e A268441 [[1]]
%e A268441 [[0], [1]]
%e A268441 [[0], [1],  [1]]
%e A268441 [[0], [1],  [3],         [1]]
%e A268441 [[0], [1],  [3, 4],      [6],          [1]]
%e A268441 [[0], [1],  [10, 5],     [15, 10],     [10],     [1]]
%e A268441 [[0], [1],  [10, 15, 6], [15, 60, 15], [45, 20], [15], [1]]
%e A268441 Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of second kind (A048993).
%t A268441 BellCoeffs[n_, k_] :=  Module[{v, r},
%t A268441 v = Table[Subscript[x,j], {j,1,n}]; (* list of variables *)
%t A268441 r = Table[Subscript[x,j]->1, {j,1,n}]; (* evaluated at 1 *)
%t A268441 MonomialList[BellY[n,k,v], v, NegativeLexicographic] /. r];
%t A268441 A268441Row[n_] := Table[BellCoeffs[n,k], {k,0,n}] // Flatten;
%t A268441 Do[Print[A268441Row[n]], {n,0,8}] (* _Peter Luschny_, Feb 08 2016 *)
%t A268441 max = 9; egf = Exp[Sum[x[k]*t^k/k!, {k, 1, max}]]; P = Table[n!* SeriesCoefficient[egf, {t, 0, n}], {n, 0, max-1}]; row[n_] := (s = Split[ Sort[{ Exponent[# /. x[_] -> x, x], #}& /@ (List @@ Expand[P[[n]]])], #1[[1]] == #2[[1]]&]; Join[{0}, #[[All, 2]]& /@ (s /. x[_] -> 1) // Flatten]); row[1] = {1}; Array[row, max] // Flatten (* _Jean-François Alcover_, Feb 08 2016 *)
%o A268441 (Sage)
%o A268441 import itertools
%o A268441 def A268441_row(n):
%o A268441     c = [bell_polynomial(n,k).coefficients() for k in (0..n)]
%o A268441     if n>0: c[0] = [0]
%o A268441     return list(itertools.chain(*c))
%o A268441 for n in range(9): print(A268441_row(n))
%Y A268441 Cf. A048993, A268442.
%K A268441 nonn,tabf
%O A268441 0,9
%A A268441 _Peter Luschny_, Feb 07 2016