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A049404 Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0). 7

%I

%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="http://www.itp.kit.edu/~wl/EISpub/A049404.text">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!.

%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-Fran├žois Alcover_, Jun 22 2018 *)

%o (Sage)

%o # The function bell_matrix is defined in 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

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Last modified February 16 12:48 EST 2019. Contains 320163 sequences. (Running on oeis4.)