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A049411 Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).

%I A049411 #27 Mar 28 2020 10:27:28
%S A049411 1,5,1,20,15,1,60,155,30,1,120,1300,575,50,1,120,9220,8775,1525,75,1,
%T A049411 0,55440,114520,36225,3325,105,1,0,277200,1315160,730345,112700,6370,
%U A049411 140,1,0,1108800,13428800,13000680,3209745,291060,11130,180,1,0,3326400
%N A049411 Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).
%C A049411 Previous name was: A triangle of numbers related to triangle A049327.
%C A049411 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 A049411 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 A049411 For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016
%H A049411 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 A049411 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%F A049411 a(n, m) = n!*A049327(n, m)/(m!*6^(n-m));
%F A049411 a(n, m) = (6*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
%F A049411 a(n, m) = 0, n<m; a(n, 0) := 0; a(1, 1) = 1.
%F A049411 E.g.f. for m-th column: (((-1+(1+x)^6)/6)^m)/m!.
%e A049411 Row polynomial E(3,x) = 20*x + 15*x^2 + x^3.
%e A049411 Triangle starts:
%e A049411 {  1}
%e A049411 {  5,    1}
%e A049411 { 20,   15,   1}
%e A049411 { 60,  155,  30,  1}
%e A049411 {120, 1300, 575, 50, 1}
%t A049411 rows = 10;
%t A049411 a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
%t A049411 Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o A049411 (Sage) # uses[bell_matrix from A264428]
%o A049411 # Adds 1,0,0,0,... as column 0 at the left side of the triangle.
%o A049411 bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # _Peter Luschny_, Jan 16 2016
%Y A049411 Cf. A049327.
%Y A049411 Row sums give A049428.
%K A049411 easy,nonn,tabl
%O A049411 1,2
%A A049411 _Wolfdieter Lang_
%E A049411 New name from _Peter Luschny_, Jan 16 2016