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A048786 Triangle of coefficients of certain exponential convolution polynomials.

%I A048786 #19 May 18 2019 17:26:26
%S A048786 1,8,1,96,24,1,1536,576,48,1,30720,15360,1920,80,1,737280,460800,
%T A048786 76800,4800,120,1,20643840,15482880,3225600,268800,10080,168,1,
%U A048786 660602880,578027520,144506880,15052800,752640,18816,224,1
%N A048786 Triangle of coefficients of certain exponential convolution polynomials.
%C A048786 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 A048786 Also the Bell transform of A034177. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%C A048786 Also the fourth power of the unsigned Lah triangular matrix A105278. - _Shuhei Tsujie_, May 18 2019
%C A048786 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 A048786 S. Roman, The Umbral Calculus, Academic Press, New York, 1984
%H A048786 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 A048786 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 A048786 Triangle begins:
%e A048786       1;
%e A048786       8,     1;
%e A048786      96,    24,    1;
%e A048786    1536,   576,   48,  1;
%e A048786   30720, 15360, 1920, 80, 1;
%e A048786   ...
%p A048786 # The function BellMatrix is defined in A264428.
%p A048786 # Adds (1,0,0,0, ..) as column 0.
%p A048786 BellMatrix(n -> 4^n*(n+1)!, 9); # _Peter Luschny_, Jan 28 2016
%t A048786 rows = 8;
%t A048786 t = Table[4^n*(n+1)!, {n, 0, rows}];
%t A048786 T[n_, k_] := BellY[n, k, t];
%t A048786 Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%Y A048786 Cf. A034177, A038231, A008297.
%K A048786 easy,nonn,tabl
%O A048786 1,2
%A A048786 _Wolfdieter Lang_
%E A048786 T(8,4) corrected by _Jean-François Alcover_, Jun 22 2018