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%I #27 Mar 28 2020 10:27:28
%S 1,5,1,20,15,1,60,155,30,1,120,1300,575,50,1,120,9220,8775,1525,75,1,
%T 0,55440,114520,36225,3325,105,1,0,277200,1315160,730345,112700,6370,
%U 140,1,0,1108800,13428800,13000680,3209745,291060,11130,180,1,0,3326400
%N Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).
%C Previous name was: A triangle of numbers related to triangle A049327.
%C a(n,1) = A008279(5,n-1). a(n,m) =: S1(-5; 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). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A013988(n,m).
%C 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 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%F a(n, m) = n!*A049327(n, m)/(m!*6^(n-m));
%F a(n, m) = (6*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: (((-1+(1+x)^6)/6)^m)/m!.
%e Row polynomial E(3,x) = 20*x + 15*x^2 + x^3.
%e Triangle starts:
%e { 1}
%e { 5, 1}
%e { 20, 15, 1}
%e { 60, 155, 30, 1}
%e {120, 1300, 575, 50, 1}
%t rows = 10;
%t a[n_, m_] := BellY[n, m, Table[k! Binomial[5, 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(5, n), 8) # _Peter Luschny_, Jan 16 2016
%Y Cf. A049327.
%Y Row sums give A049428.
%K easy,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
%E New name from _Peter Luschny_, Jan 16 2016
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