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

1,2

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

References

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

Links

Table of n, a(n) for n=1..36.

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

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).

Example

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

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 *)

Crossrefs

Cf. A034177, A038231, A008297.

Sequence in context: A347111 A254933 A174503 * A240955 A132056 A051187

Adjacent sequences: A048783 A048784 A048785 * A048787 A048788 A048789

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Extensions

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

Status

approved