%I #19 May 18 2019 17:26:26
%S 1,8,1,96,24,1,1536,576,48,1,30720,15360,1920,80,1,737280,460800,
%T 76800,4800,120,1,20643840,15482880,3225600,268800,10080,168,1,
%U 660602880,578027520,144506880,15052800,752640,18816,224,1
%N Triangle of coefficients of certain exponential convolution polynomials.
%C i) p(n,x) := sum(a(n,m)*x^m,m=1..n), p(0,x) := 1, are monic polynomials satisfying p(n,x+y)= sum(binomial(n,k)*p(k,x)*p(n-k,y),k=0..n), (exponential convolution polynomials). ii) In the terminology of the umbral calculus (see reference) p(n,x) are called associated to f(t)= t/(1+4*t). iii) a(n,1)= A034177(n).
%C Also the Bell transform of A034177. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%C Also the fourth power of the unsigned Lah triangular matrix A105278. - _Shuhei Tsujie_, May 18 2019
%C Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -3 <= d <= 4). - _Shuhei Tsujie_, May 18 2019
%D S. Roman, The Umbral Calculus, Academic Press, New York, 1984
%H N. Nakashima and S. Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%F a(n, m) = n!*4^(n-m)*binomial(n-1, m-1)/m!, n >= m >= 1; a(n, m) := 0, m>n; a(n, m) = (n!/m!)*A038231(n-1, m-1) = 4^(n-m)*A008297(n, m) (Lah-triangle).
%e Triangle begins:
%e 1;
%e 8, 1;
%e 96, 24, 1;
%e 1536, 576, 48, 1;
%e 30720, 15360, 1920, 80, 1;
%e ...
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> 4^n*(n+1)!, 9); # _Peter Luschny_, Jan 28 2016
%t rows = 8;
%t t = Table[4^n*(n+1)!, {n, 0, rows}];
%t T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%Y Cf. A034177, A038231, A008297.
%K easy,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
%E T(8,4) corrected by _Jean-François Alcover_, Jun 22 2018