A048786 Triangle of coefficients of certain exponential convolution polynomials.

1, 8, 1, 96, 24, 1, 1536, 576, 48, 1, 30720, 15360, 1920, 80, 1, 737280, 460800, 76800, 4800, 120, 1, 20643840, 15482880, 3225600, 268800, 10080, 168, 1, 660602880, 578027520, 144506880, 15052800, 752640, 18816, 224, 1

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• tabl

Comments

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).

Also the Bell transform of A034177. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

Also the fourth power of the unsigned Lah triangular matrix A105278. - Shuhei Tsujie, May 18 2019

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

Examples

			Triangle begins:
      1;
      8,     1;
     96,    24,    1;
   1536,   576,   48,  1;
  30720, 15360, 1920, 80, 1;
  ...
		

References

• S. Roman, The Umbral Calculus, Academic Press, New York, 1984

Links

• N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Crossrefs

Cf. A034177, A038231, A008297.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 4^n*(n+1)!, 9); # Peter Luschny, Jan 28 2016

Mathematica

rows = 8;
t = Table[4^n*(n+1)!, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

Formula

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).

Extensions

T(8,4) corrected by Jean-François Alcover, Jun 22 2018