%I #52 Aug 23 2023 10:45:03
%S 1,1,1,0,3,1,0,3,6,1,0,0,15,10,1,0,0,15,45,15,1,0,0,0,105,105,21,1,0,
%T 0,0,105,420,210,28,1,0,0,0,0,945,1260,378,36,1,0,0,0,0,945,4725,3150,
%U 630,45,1,0,0,0,0,0,10395,17325,6930,990,55,1,0,0,0,0,0,10395,62370
%N A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).
%C a(n,1) = A019590(n) = A008279(1,n). a(n,m) =: S1(-1; 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))*A001497(n-1,m-1) (signed Bessel triangle). 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 Exponential Riordan array [1+x, x(1+x/2)]. T(n,k) = A001498(k+1, n-k). - _Paul Barry_, Jan 15 2009
%H G. C. Greubel, <a href="/A049403/b049403.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Wolfdieter 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 Wolfdieter Lang, <a href="/A049403/a049403.txt">First 10 rows of the array and more.</a>
%H Robert S. Maier, <a href="https://arxiv.org/abs/2308.10332">Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers</a>, arXiv:2308.10332 [math.CO], 2023. See p. 19.
%F a(n, m) = n!*A030528(n, m)/(m!*2^(n-m)) for n >= m >= 1.
%F a(n, m) = (2*m-n+1)*a(n-1, m) + a(n-1, m-1) for n >= m >= 1 with a(n, m) = 0 for n < m, a(n, 0) := 0, and a(1, 1) = 1. [The 0th column does not appear in this array. - _Petros Hadjicostas_, Oct 28 2019]
%F E.g.f. for the m-th column: (x*(1 + x/2))^m/m!.
%F a(n,m) = A122848(n,m). - _R. J. Mathar_, Jan 14 2011
%e Triangle a(n,m) (with rows n >= 1 and columns m >= 1) begins as follows:
%e 1; with row polynomial E(1,x) = x;
%e 1, 1; with row polynomial E(2,x) = x^2 + x;
%e 0, 3, 1; with row polynomial E(3,x) = 3*x^2 + x^3;
%e 0, 3, 6, 1; with row polynomial E(4,x) = 3*x^2 + 6*x^3 + x^4;
%e 0, 0, 15, 10, 1;
%e 0, 0, 15, 45, 15, 1;
%e 0, 0, 0, 105, 105, 21, 1;
%e 0, 0, 0, 105, 420, 210, 28, 1;
%e ...
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> `if`(n<2,1,0), 9); # _Peter Luschny_, Jan 28 2016
%t t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten
%t (* Second program: *)
%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
%t rows = 13;
%t M = BellMatrix[If[#<2, 1, 0]&, rows];
%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%Y Cf. A000085 (row sums), A001497, A001498, A008275, A008279, A008297, A019590, A030528, A039692.
%Y Variations of this array: A096713, A104556, A122848, A130757.
%K easy,nonn,tabl
%O 1,5
%A _Wolfdieter Lang_