%I #35 Mar 27 2020 06:58:50
%S 1,2,1,2,6,1,0,20,12,1,0,40,80,20,1,0,40,360,220,30,1,0,0,1120,1680,
%T 490,42,1,0,0,2240,9520,5600,952,56,1,0,0,2240,40320,48720,15120,1680,
%U 72,1,0,0,0,123200,332640,184800,35280,2760,90,1,0,0,0,246400,1786400
%N Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).
%C Previous name was: A triangle of numbers related to triangle A049324.
%C a(n,1) = A008279(2,n-1). a(n,m) =: S1(-2; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers).
%C a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A004747(n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%C For the definition of the Bell transform see A264428 and the link. - _Peter Luschny_, Jan 16 2016
%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H W. Lang, <a href="/A049404/a049404.txt">First 10 rows of the array and more.</a> [From _Wolfdieter Lang_, Oct 17 2008]
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%F a(n, m) = n!*A049324(n, m)/(m!*3^(n-m));
%F a(n, m) = (3*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
%F a(n, m) = 0, n<m; a(n, 0) = 0; a(1, 1) = 1.
%F E.g.f. for m-th column: ((x+x^2+(x^3)/3)^m)/m!.
%F a(n,m) = n!/(3^m * m!)*(Sum_{i=0..floor(m-n/3)} (-1)^i * binomial(m,i) * binomial(3*m-3*i,n)), 0 for empty sums. - _Werner Schulte_, Feb 20 2020
%e E.g. row polynomial E(3,x) = 2*x+6*x^2+x^3.
%e Triangle starts:
%e {1}
%e {2, 1}
%e {2, 6, 1}
%e {0, 20, 12, 1}
%t rows = 11;
%t a[n_, m_] := BellY[n, m, Table[k! Binomial[2, k], {k, 0, rows}]];
%t Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o (Sage) # uses[bell_matrix from A264428]
%o # Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
%o bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # _Peter Luschny_, Jan 16 2016
%Y Row sums give A049425.
%Y Cf. A004747, A049324.
%K easy,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
%E New name from _Peter Luschny_, Jan 16 2016