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A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).

%I A049424 #27 Mar 28 2020 10:58:18
%S A049424 1,4,1,12,12,1,24,96,24,1,24,600,360,40,1,0,3024,4200,960,60,1,0,
%T A049424 12096,40824,17640,2100,84,1,0,36288,338688,270144,55440,4032,112,1,0,
%U A049424 72576,2407104,3580416,1212624,144144,7056,144,1,0,72576,14515200,41791680
%N A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).
%C A049424 Previous name was: A triangle of numbers related to triangle A049326.
%C A049424 a(n,1) = A008279(4,n-1). a(n,m) =: S1(-4; 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))*A011801(n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%C A049424 For the definition of the Bell transform see A264428 and the link. - _Peter Luschny_, Jan 16 2016
%H A049424 W. Lang, <a href="https://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 A049424 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%F A049424 a(n, m) = n!*A049326(n, m)/(m!*5^(n-m));
%F A049424 a(n, m) = (5*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
%F A049424 a(n, m) = 0, n<m; a(n, 0) := 0; a(1, 1) = 1.
%F A049424 E.g.f. for m-th column: (((-1+(1+x)^5)/5)^m)/m!.
%e A049424 E.g., row polynomial E(3,x) = 12*x + 12*x^2 + x^3.
%e A049424 Triangle starts:
%e A049424    1;
%e A049424    4,   1;
%e A049424   12,  12,   1;
%e A049424   24,  96,  24,   1;
%e A049424   24, 600, 360,  40,   1;
%t A049424 rows = 10;
%t A049424 a[n_, m_] := BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
%t A049424 Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o A049424 (Sage) # uses[bell_matrix from A264428]
%o A049424 # Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
%o A049424 bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # _Peter Luschny_, Jan 16 2016
%Y A049424 Cf. A049326.
%Y A049424 Row sums give A049427.
%K A049424 easy,nonn,tabl
%O A049424 1,2
%A A049424 _Wolfdieter Lang_
%E A049424 New name from _Peter Luschny_, Jan 16 2016