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%I #19 Mar 26 2020 03:56:57
%S 1,3,1,6,9,1,6,51,18,1,0,210,195,30,1,0,630,1575,525,45,1,0,1260,
%T 10080,6825,1155,63,1,0,1260,51660,71505,21840,2226,84,1,0,0,207900,
%U 623700,333585,57456,3906,108,1,0,0,623700,4573800,4293135,1195425,131670
%N A triangle of numbers related to triangle A049325.
%C a(n,1)= A008279(3,n-1). a(n,m)=: S1(-3; 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))*A000369(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 Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+3) (A008545) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - _Peter Luschny_, Dec 31 2015
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F a(n, m) = n!*A049325(n, m)/(m!*4^(n-m)); a(n, m) = (4*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n<m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: (((-1+(1+x)^4)/4)^m)/m!.
%e Triangle begins:
%e {1};
%e {3,1};
%e {6,9,1};
%e {6,51,18,1};
%e ...
%e E.g. row polynomial E(3,x)= 6*x+9*x^2+x^3.
%t rows = 10;
%t t = Table[Product[4k+3, {k, 0, n-1}], {n, 0, rows}];
%t T[n_, k_] := BellY[n, k, t];
%t M = Inverse[Array[T, {rows, rows}]] // Abs;
%t A049325 = Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o (Sage) # uses[inverse_bell_transform from A265605]
%o # Adds a column 1,0,0,0,... at the left side of the triangle.
%o multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))
%o inverse_bell_matrix(multifact_4_3, 9) # _Peter Luschny_, Dec 31 2015
%Y Row sums give A049426.
%K easy,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
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